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Question: <p>In a proof I found in a book regarding the ODE <span class="math-container">$\dot{x}(t) = A(t)x(t)$</span> where <span class="math-container">$A$</span> is piecewise continuous on a compact interval I, it is assumed that the maximum <span class="math-container">$K = \max_I ||A(t)||$</span> exists. In my op... | https://math.stackexchange.com/questions/2973787/why-can-we-assume-that-a-piecewise-continuous-function-has-a-maximum |
Question: <p>I have been attacking this question but i got stuck. Please give me some hint.</p>
<p>Let $1\leq p<\infty$ and $\{f_n\}\subset L^p$. Suppose $f_n\to f$ in measure and $\|f_n\|_p\to\|f\|_p$, then $f_n\to f$ in $L^p$ norm.</p>
Answer: <p>It's not easy. The key idea is to mimic the proof of Dominated Con... | https://math.stackexchange.com/questions/1162919/convergence-in-measure-and-convergence-of-norm-implies-convergence-in-lp |
Question: <blockquote>
<p>It's the follow metric: <span class="math-container">$d(x,y)= ||x|| +||y||$</span> if <span class="math-container">$x$</span> and <span class="math-container">$y$</span> don't lie on a line through the origin. And otherwise <span class="math-container">$d(x,y)= ||x-y||$</span>.</p>
<p>I think ... | https://math.stackexchange.com/questions/1319739/is-mathbbr2-separable-for-the-railmetric-or-sncf-metric-or-post-office |
Question: <p>Let <span class="math-container">$$f(x) =
\begin{cases}
\dfrac{1}{q^2}, & \text{if $x=\dfrac{p}{q} $ is rational and in lowest terms;} \\[2ex]
0, & \text{if $x$ is irrational}
\end{cases}$$</span>
Where is f continuous? Is f differentiable anywhere?</p>
<p>My attempt: I can prove that <span class=... | https://math.stackexchange.com/questions/4178091/a-question-on-differentiability-of-a-variant-of-thomae-function |
Question: <p>Suppose that <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are differentiable functions on an open interval <span class="math-container">$I$</span> and that <span class="math-container">$p \in I$</span>. If <span class="math-container">$\lim_{x \to p} f(x) = \lim_{x \t... | https://math.stackexchange.com/questions/4178423/ask-a-step-in-proof-of-lhospitals-rule |
Question: <p>Let <span class="math-container">$f \in C[0,1]$</span>. If <span class="math-container">$\{y_i\} \subset f([0,1])$</span> is increasing, there exists an monotonous <span class="math-container">$\{x_i\} \subset [0,1]$</span> such that <span class="math-container">$f(x_i)=y_i$</span>?</p>
<p>Intuitively, fro... | https://math.stackexchange.com/questions/4181505/if-y-i-subset-f0-1-is-increasing-there-exists-an-monotonous-x-i |
Question: <p>***How to find the this limit without using L'Hopital's Rule.</p>
<p><span class="math-container">$\lim _{x\to 0^+}\frac{\left(e^x-e^{-x}\right)}{\sin x}$</span></p>
Answer: <p>Hint:</p>
<ul>
<li><p>Use maclaurin series for <span class="math-container">$e^x$</span> and <span class="math-container">$e^{-x}... | https://math.stackexchange.com/questions/3982661/calculate-lim-x-to-0-frac-leftex-e-x-right-sin-x |
Question: <p>Hi I was wondering if it were possible to skip undergraduate and apply to a graduate school. I have been taking some math courses through CTY a somewhat accredited program offered by Johns Hopkins university. They offer up to real analysis 1 and complex analysis 1. If I take up to there, would it be possib... | https://math.stackexchange.com/questions/4177654/would-it-be-possible-to-skip-undergraduate-college-with-self-study-online-course |
Question: <p><span class="math-container">$$\lim_{n\to\infty}\left(1+\dfrac{1}{n+1}\right) ^{n+1} =e$$</span></p>
<p>Could someone explain me why this is trivial ? Maybe subsitution such that <span class="math-container">$n+1=m$</span> which would give us the definition of <span class="math-container">$e$</span> ? </p... | https://math.stackexchange.com/questions/3115588/why-is-lim-limits-n-to-infty-left1-dfrac1n1-rightn1-e-trivial |
Question: <p>I want to formally prove that this limit is equal to <span class="math-container">$0$</span></p>
<p><span class="math-container">$\lim_{x \to 0} \frac{2}{x^{3}} e^{-\frac{1}{x^{2}}}$</span></p>
<p>can anybody help me?</p>
Answer: <p>This is equivalent to</p>
<p><span class="math-container">$$\lim_{t\to... | https://math.stackexchange.com/questions/3705034/resolution-of-a-limit |
Question: <p>The definition of Thomae function is
<span class="math-container">$$g(x) =
\begin{cases}
\dfrac{1}{q}, & \text{if $x=\dfrac{p}{q}$} \\[2ex]
0, & \text{if $x$ is irrational}
\end{cases}$$</span>
where <span class="math-container">$p\in \mathbb{Z}, q\in \mathbb{Z}^+$</span> and <span class="math-con... | https://math.stackexchange.com/questions/4189450/a-question-on-the-riemann-integrable-of-thomae-function |
Question: <p>My book says that if functions $f$ and $g$ are both onto then $f\circ g$ and $g\circ f$ may or may not be onto. </p>
<p>Why is this so? Would someone please help me understand this, maybe with an example or diagrammatically? My book states that$ f\circ g$ and $g\circ f$ may or may not be onto. I think th... | https://math.stackexchange.com/questions/2913101/explanation-of-composition-of-two-onto-functions |
Question: <p>I'm working on the following question:</p>
<blockquote>
<p>Which of the following three properties imply <span class="math-container">$f$</span> is monotone (it maybe none or more than 1):</p>
<ol>
<li><span class="math-container">$f(x + y) = f(x)f(y)$</span></li>
<li><span class="math-containe... | https://math.stackexchange.com/questions/2935065/possible-properties-of-a-monotone-function |
Question: <p>I'm working on the following problem:</p>
<blockquote>
<p>A real valued function <span class="math-container">$f$</span> defined on <span class="math-container">$\mathbb{R}$</span> has the property that <span class="math-container">$(\forall \epsilon>0)(\exists\delta>0)$</span> s.t. <span class="m... | https://math.stackexchange.com/questions/2937282/epsilon-delta-exercise |
Question: <p>The radius of convergence <span class="math-container">$r$</span> can be calculated for every power series <span class="math-container">$\sum_{k=0}^\infty a_k z^k$</span> with <span class="math-container">$a_k\in \mathbb C$</span> and <span class="math-container">$z\in \mathbb C$</span> by using the Cauchy... | https://math.stackexchange.com/questions/2941938/cauchy-hadamard-formula-and-starting-index-of-power-series |
Question: <p>How can i prove uniform convergence on <span class="math-container">$E=[0, \frac{\pi}{2}]$</span> ?
<span class="math-container">$$\sum^\infty _{n=1} {x e^{-nx}\cos(nx)}$$</span></p>
Answer: <p>Hint: For <span class="math-container">$N$</span> even, let</p>
<p><span class="math-container">$$T_N(x)=\sum_{... | https://math.stackexchange.com/questions/2962462/uniform-convergence-of-sum-infty-n-1-x-e-nx-cosnx |
Question: <p>Consider a sequence <span class="math-container">$x_n$</span> with positive values such that
<span class="math-container">$\sum _{n=1}^{\infty} x_n $</span> converges .
Is the set of the subsequences <span class="math-container">$ x_{k_n}$</span> of <span class="math-container">$x_n$</span> such that
<sp... | https://math.stackexchange.com/questions/3035548/countability-of-a-set-of-subsequences |
Question: <p>I realize there are several questions dealing with Theorem 1.1 from Apostol's "Mathematical Analysis", but those questions are mostly about how to prove the result or the geometrical intuition around it. My question is about the application of the theorem. The theorem states:</p>
<p>If <span class="math-c... | https://math.stackexchange.com/questions/3039044/apostol-mathematical-analysis-theorem-1-1 |
Question: <p>I am working on exercice 9.5.2 of Analysis by Zorich and I am stuck at the question b.</p>
<blockquote>
<p>a) A set <span class="math-container">$E\subset X$</span> of a metric space <span class="math-container">$(X,d)$</span> is nowhere dense in X if it is not dense in any ball, that is, if for every ball... | https://math.stackexchange.com/questions/3042725/if-at-each-point-of-a-closed-interval-the-m-th-derivative-of-f-is-0-for-m |
Question: <p>Show that <span class="math-container">$\left( x_{n}\right) $</span> is a Cauchy sequence, where
<span class="math-container">$$
x_{n}=\frac{\sin1}{2}+\frac{\sin2}{2^{2}}+\ldots+\frac{\sin n}{2^{n}}.
$$</span></p>
<p>We try to evaluate <span class="math-container">$\left\vert x_{n+p}-x_{n}\right\vert $<... | https://math.stackexchange.com/questions/3061670/proof-that-a-sequence-is-cauchy |
Question: <p>Why is Completeness needed to demonstrate the Archimedean principle?</p>
<p>Could someone criticize the following proof.</p>
<p>Thanks in advance.</p>
<hr>
<p><strong>Proof</strong></p>
<p>Considering any <span class="math-container">$x \in \mathbb{R},$</span> <span class="math-container">$\lfloor{x}\... | https://math.stackexchange.com/questions/3099533/why-is-completeness-needed-to-demonstrate-the-archimedean-principle |
Question: <p>We already know that
<span class="math-container">$$\lim_{n \rightarrow +\infty} \int_{-1}^1 (1-x^2)^n dx = 0$$</span></p>
<p>If we have <span class="math-container">$f(x) \in C[-1,1]$</span> then prove
<span class="math-container">$$\lim_{n \rightarrow +\infty} \frac{\int_{-1}^1 f(x)(1-x^2)^n dx}{\int_{-... | https://math.stackexchange.com/questions/3118986/frac-int-fg-dx-int-g-dx-f0 |
Question: <p><strong>Problem</strong></p>
<blockquote>
<p>Let <span class="math-container">$u(x,t) = \frac{e^\frac{-x^2}{4t}}{\sqrt{4 \pi t}} $</span> for <span class="math-container">$ t > 0,
> x \in \mathbb{R} $</span>. If a > 0, prove that <span class="math-container">$u(x,t) \rightarrow 0$</span> as <span ... | https://math.stackexchange.com/questions/3155369/uniformly-convergence-of-heat-equation |
Question: <p>When we shift <span class="math-container">$\sin(x)$</span> by <span class="math-container">$\pi$</span> (half the period) we get an odd function <span class="math-container">$-\sin(x)$</span>.
I was wondering if every periodic, even function can be made odd if we shift it by half it's period?</p>
<p>I gu... | https://math.stackexchange.com/questions/3207241/shift-of-even-periodic-function |
Question: <p>I am trying to read <em>Mathematical Analysis I</em> by Zorich on my own. Here is an exercise that I could not solve: </p>
<blockquote>
<p>For all <span class="math-container">$l \in \mathbb{R}$</span> that is not of the form <span class="math-container">$\frac{1}{n}$</span> for some <span class="math-c... | https://math.stackexchange.com/questions/3230799/a-problem-from-zorich |
Question: <p>Let's say a function <span class="math-container">$f:[a,b] \to \mathbb{R}$</span> is bounded and Riemann integrable, then would there always exist a partition <span class="math-container">$P$</span> of <span class="math-container">$[a,b]$</span> such that the lower sum of <span class="math-container">$P$<... | https://math.stackexchange.com/questions/3275494/riemann-integrability-and-lower-and-upper-sums |
Question: <p>I'm lost with the assignment from my teacher.
For real numbers w, x, y, and z. Prove for x,z≠0, (w/x)(y/z)=(wy)/(xz) and (w/x)+(y/z)=(wz+xy)/(xz)</p>
Answer: <p>You can write <span class="math-container">$$\frac{w}{x}+\frac{y}{z}=\frac{wz}{xz}+\frac{xy}{xz}=…$$</span>
<span class="math-container">$$\frac... | https://math.stackexchange.com/questions/3355313/for-real-numbers-w-x-y-and-z-prove-for-x-z-%e2%89%a0-0-w-xy-z-wy-xz-and-w |
Question: <blockquote>
<p>Suppose <span class="math-container">$f(x)$</span> defines on the bounded interval <span class="math-container">$I$</span>. Prove that <span class="math-container">$f(x)$</span> is uniformly continuous on <span class="math-container">$I$</span> if and only if the image of each Cauchy sequenc... | https://math.stackexchange.com/questions/3381457/prove-that-fx-is-uniformly-continuous-on-i-if-and-only-if-the-image-of-eac |
Question: <blockquote>
<p>Suppose <span class="math-container">$f(x)$</span> is differentiable on <span class="math-container">$[0,\,1]$</span>, <span class="math-container">$f(0)=0$</span>, <span class="math-container">$f(1)=1$</span> and <span class="math-container">$p_1,\,p_2,\cdots,\,p_n$</span> are <span class="... | https://math.stackexchange.com/questions/3383736/prove-there-are-distinct-x-1-x-2-cdots-x-n-such-that-sum-i-1n-frac |
Question: <p>Find the explicit form of
<span class="math-container">$$
\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n(n+2)}x^{n-1}.
$$</span></p>
<p>Let <span class="math-container">$S(x)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n(n+2)}x^{n-1}$</span>. It has radius of convergence <span class="math-container">$1$</span>.</p>
<p>... | https://math.stackexchange.com/questions/3386371/find-the-explicit-form-of-sum-n-1-infty-frac-1n-1nn2xn-1 |
Question: <blockquote>
<p>Suppose <span class="math-container">$\displaystyle f(x)=\sum_{n=0}^{\infty}2^{-n}\sin(2^nx)$</span>, evaluate
<span class="math-container">$$
\lim_{x\to0+}\frac{f(x)-f(0)}{x}.
$$</span></p>
</blockquote>
<p><span class="math-container">$f(x)$</span> converges uniformly on <span class="ma... | https://math.stackexchange.com/questions/3401237/evaluate-lim-x-to0-sum-n-0-infty2-n-frac-sin-2nxx |
Question: <p>So I have to prove if is <span class="math-container">$\mathbb{N} \times \mathbb{R}$</span> closed, open or neither?</p>
<p>Any help?</p>
Answer: <p>In what topological space? If <span class="math-container">$\mathbf{R}^2$</span>, closed. If <span class="math-container">$\mathbf{R}\times\mathbf{N}$</spa... | https://math.stackexchange.com/questions/3448363/is-mathbbn-times-mathbbr-closed-open-or-neither |
Question: <p><span class="math-container">$f(x)$</span> is differentiable, and for any <span class="math-container">$x\in \mathbb R$</span>, <span class="math-container">$|f'(x)|\le \lambda |x|$</span>, then how to show <span class="math-container">$f\equiv 0$</span> ? </p>
<p>This is a question my student ask me, ... | https://math.stackexchange.com/questions/3457881/how-to-show-f-equiv-0 |
Question: <p>Assume <span class="math-container">$f\in C^2(D)$</span>, where <span class="math-container">$D=\{(x,y)\in \mathbb R^2: x^2+ y^2 \le 1\}$</span>, if
<span class="math-container">$$
\frac{\partial^2 f }{\partial x^2} + \frac{\partial^2 f }{\partial y^2}
=e^{-x^2 -y^2}
$$</span>
how do I show
<span class="... | https://math.stackexchange.com/questions/3465278/how-to-show-iint-limits-dx-frac-partial-f-partial-x-y-frac-partial-f |
Question: <p>Let <span class="math-container">$f(x)$</span> be a continuous function on <span class="math-container">$[a, b]$</span> (not necessarily derivable) and satisfy <span class="math-container">$f(a)<f(b)$</span>. At the same time, the limit <span class="math-container">$g(x)=\lim_{t \to 0} \frac{f(x+t)-f(x... | https://math.stackexchange.com/questions/3544900/gx-lim-t-to-0-fracfxt-fx-tt-prove-that-there-is-a-c-ina-b |
Question: <p>Consider the non-negative orthant
<span class="math-container">$$
R\equiv \{(x_1,...,x_n): x_i\geq 0 \text{ }\forall i\}
$$</span>
The boundary of <span class="math-container">$R$</span> is not smooth.
I'm looking for a function which can smooth the boundary of <span class="math-container">$R$</span> and w... | https://math.stackexchange.com/questions/3606304/non-negative-orthant-smoothing |
Question: <p>I picked up a copy of Principles of Mathematical Analysis by Rudin. It just so happened to be the second edition which is drastically different from the third and I can't find solutions to a lot of exercises.</p>
<p>I'd like some hints on how to complete the following few problems.</p>
<ol>
<li>If <span ... | https://math.stackexchange.com/questions/3647631/few-questions-1-4-1-5-1-6-from-rudin-2e-on-roots-of-reals |
Question: <p>The limit location theorem states for a sequence $\{a_n\}$, if for large enough $n$ we have $a_n \leq M$, then $\lim_{n \to \infty} a_n \leq M$ (the same holds if we replace $\leq$ with $\geq$). Why does the theorem not hold with strict inequalities?</p>
Answer: <p>In general we have that even if $a_n <... | https://math.stackexchange.com/questions/2784352/why-cant-we-replace-leq-with-in-the-limit-location-theorem |
Question: <p>For $N\geq 1$ the discrete Fourier Coefficient of $f$ are defined by
\begin{align}
a_N(n)=\frac{1}{N}\sum_{k=1}^{N}f(e^{2\pi i k/N})e^{-2\pi i k n/N}, \quad \forall n\in \mathbb{Z}
\end{align}
Suppose $f\in C^1$ function on circle, how to prove there exist $c>0$ such that $|a_N(n)|\leq \frac{c}{|n|}$ wh... | https://math.stackexchange.com/questions/2750867/a-nn-leq-fraccn |
Question: <p>Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ defined by \begin{equation*}f: \begin{pmatrix}x \\ y\end{pmatrix}\rightarrow \begin{pmatrix}u \\ v\end{pmatrix}=\begin{pmatrix}x(1-y) \\ xy\end{pmatrix}\end{equation*} </p>
<p>I want to determine th eimage $f(\mathbb{R}^2)$. </p>
<p>$$$$ </p>
<p>We have that ... | https://math.stackexchange.com/questions/2822614/determine-the-image-of-the-map |
Question: <p>I am relevantly new to this concept of Bounded variation.
I want to know is the function $\sqrt {1-x^2} $, $x\in (-1,1) $ of bounded variation? How should I proceed. Is there any graphical interpretation of functions of bounded variation?</p>
Answer: <p>As for the specific question, $f(x)=\sqrt{1-x^2}$ i... | https://math.stackexchange.com/questions/2823531/is-this-function-a-function-of-bounded-variation |
Question: <p>$f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and $K\subset\mathbb{R}$ is compact.<br>
$\{Q_i : i\in I\}$ is an open cover for $f(K)$.<br>
Does it follow that $U_i:= f^{-1}(O_i)$ is open for each $i \in I$? Is $\{U_i:i\in I \}$ an open cover for $K$?<br>
Intuitively, the answer to both of these questi... | https://math.stackexchange.com/questions/2791505/q-i-i-in-i-text-is-open-rightarrow-u-i-f-1o-i-text-is-open |
Question: <p>So I have a problem that's asking me to compute the intersection of all sets $S(k)$ such that</p>
<p>$$ S(k) = \left(\frac{k-1}{k}, \frac{2k+1}{k}\right], k \ge 1 $$</p>
<p>My answer is $(1,2)$, but the book's answer is $(0,2)$. Maybe I'm just dumb, but I do not understand why the book's answer would be ... | https://math.stackexchange.com/questions/2824115/trouble-with-intersection-of-infinitely-many-sets |
Question: <p>I've got a task, where a set A and a map g were given.
The task was to calculate g(A).
And I don't know what to do exactly, so I'd appreciate if s.o. could explain it with an example or give me a link, where it's explained.</p>
<p>For example let A = { x in IR | |x-3|= 2 } and g (x) = (x+2)/x.</p>
<p>So ... | https://math.stackexchange.com/questions/1530276/how-do-i-calculate-a-set-in-a-map |
Question: <p>I am reading a paper on Markov chains and am trying to prove a lemma left to the reader that I need for the Ergodic Theorem for Markov Chains (though the lemma requires no knowledge of any of this). The statement of the lemma is as follows:</p>
<p>Let $\{a_n\}$ be a bounded sequence and suppose for $\{n_k... | https://math.stackexchange.com/questions/1521861/for-a-bounded-sequence-a-n-lim-limits-k-to-inftya-1-a-n-k-n-k-a |
Question: <p><strong>EDIT:</strong> Sorry. I basically was confused by that just valid mathematical transforming could lead into a undefined behavior.
I have to admit, my question has not much to do with the zero of a function, i've just used one method of the "zero calculating tool" set to reshape my function. (i exp... | https://math.stackexchange.com/questions/1536458/transforming-x3-x2-1-implies-x-leftx2-x-dfrac1x-right-imp |
Question: <p>Would this even assist math in the way that $i$ did? Or is this just outright pointless and/or too exclusive to call for a definition?</p>
Answer: <p>It would require us to give up many of the usual rules of algebra -- for example the cancellation rule which says that
$$ \frac{a\times b}{a} = b $$
wheneve... | https://math.stackexchange.com/questions/1558737/if-i2-1-at-one-point-undefined-then-why-not-define-frac00 |
Question: <p>Let $f \colon [a, b] →\mathbb R$ be a continuous function. Then prove that the graph of $f$,
$$\operatorname{Graph}(f) := \{\,(y, x) \in \mathbb R^2\mid y = f(x), x \in [a, b]\,\}$$ is a Jordan region, and it has Jordan content $0$.</p>
<p>So I need to show that there exist an $\epsilon >0$ such that ... | https://math.stackexchange.com/questions/1374570/show-the-graph-is-jordan-region-with-volume-0 |
Question: <p>Is there an infinite closed subset $X$ of the unit circle in $\mathbb C$ such that the squaring map induces a bijection from $X$ to itself?</p>
Answer: <p>Let us think of $S^1$ as $\mathbb{R}/\mathbb{Z}$, so we want an infinite closed subset $X$ on which multiplication by $2$ is a bijection. Suppose you ... | https://math.stackexchange.com/questions/1539215/infinite-closed-subset-of-s1-such-that-the-squaring-map-is-a-bijection |
Question: <p>Here's an exercise in normed spaces that I can't get my head around.
It reads as follows:
"Let X be a compact space equipped with norm d. If X is countable, then the set of isolated points in X is both open and dense."
Just point me in the right direction.
Thanks a bunch!</p>
Answer: <p>You've already fi... | https://math.stackexchange.com/questions/1596301/isolated-points-within-a-compact-space |
Question: <p>Let $S= \{(x,y): y<2x+1\}$.Then prove that $S$ is open in $\mathbb{R}\times\mathbb{R}$.</p>
<p>To prove that we must show that</p>
<blockquote>
<p>for every $(u,v)\in S $ there exists $r>0$ such that the ball $B[(u,v),r]$ is contained in $S$</p>
</blockquote>
<p>Or, equivalently,</p>
<blockquot... | https://math.stackexchange.com/questions/1539580/prove-that-a-particular-set-is-open |
Question: <p>How to write this in math notation
$$A=\{2,4,...,2n,...\}
\\B=\{3,6,9,...,3n,...\}$$
what is $A\cup B$ and $A\setminus B$</p>
<p>by inspection, I can see that $A\cup B=\{2,3,4,6,8,9...\}=\{2,4,6...3,6,9...\}$ that is even set or multiple of 3 but I do not know how to write this in math notation
$ A\setmin... | https://math.stackexchange.com/questions/1457643/set-operation-for-union-and-intersection |
Question: <p>$f(x)=x^2\sin^2(1/x)$ for $x\neq0$ and $f(0)=0$ for $x=0$</p>
<p>How to show $f(x)$ is Lipschitz continuous on $[-a,a]$.</p>
<p>I tried to use Mean Value Theorem, but couldn't finalized it.</p>
Answer: <p>If you want to find an $K > 0$ such that $|f'(x)| \leq K$, then $K$ is not that difficult to cal... | https://math.stackexchange.com/questions/1598200/how-to-show-fx-x2-sin21-x-is-lipschitz-continuous |
Question: <p>I am trying to prove that: Consider the sequence $a_n = \sup_{x\in S}|f_n(x) - f(x)|$. Then $f_n$ converges to $f$ uniformly if and only if $a_n$ tends to $0$. But I can't prove that if $f_n$ converges to $f$ uniformly, then $a_n$ tends to $0$. I have not been able to go from $|f_n(x) - f(x)|&l... | https://math.stackexchange.com/questions/1276263/uniform-convergence-and-maximum-of-an-absolute-difference |
Question: <p>Let $T$ be a nonzero linear functional on a normed space $X$.
Is it then true that on each ball $B(x,r)$, $f$ is not constant?</p>
Answer: <p>Sure, you can consider for the sake of clarity $B(0,r)$. Take $x_1$ and $x_2$ such that $||x_1||,||x_2|| < \frac{r}{4}$ and $T(x_1) \neq 0$ $T(x_2) \neq 0$ (the... | https://math.stackexchange.com/questions/1564480/is-a-nonzero-linear-functional-is-not-constant-on-each-ball |
Question: <p>How to find $f(\mathbb{R}^2)$ if $f:\mathbb{R}^2 \to \mathbb{R}^2$</p>
<p>$$f(x,y)=(e^{x+y}+e^{x-y},e^{x+y}-e^{x-y} )$$</p>
Answer: | https://math.stackexchange.com/questions/1609597/range-of-a-function-of-two-variable |
Question: <blockquote>
<p>Prove that there exists a smallest positive number $p$ such that $\cos(p) = 0
$.</p>
</blockquote>
<p>I think I'm supposed to use either Rolle's theorem or the Mean Value Theorem but I'm not sure how, any help would be appreciated. Thanks!</p>
Answer: <p>$\cos$ is a continous function, th... | https://math.stackexchange.com/questions/2155270/analysis-prove-that-there-exists-a-smallest-positive-number-p-such-that-co |
Question: <p>Let <span class="math-container">$z$</span> be an upper bound for <span class="math-container">$A \subset \Re$</span> such that for all <span class="math-container">$\epsilon > 0$</span> there is an <span class="math-container">$x \in A$</span> such that <span class="math-container">$x > z − \epsilon... | https://math.stackexchange.com/questions/4993999/let-z-be-an-upper-bound-for-a-subset-re-such-that-for-all-epsilon-0-t |
Question: <p>I am not sure if this is true, but intuitively it seems that if a function is strictly increasing and it is also continuous...it is differentiable. </p>
<p>It may be because there are no bumps like in the absolute value. </p>
Answer: <p>Not necessarily. Counterexample:
$$
f(x)=\begin{cases}
x & \te... | https://math.stackexchange.com/questions/130457/continuous-and-strictly-increasing-implies-differentiable |
Question: <p>All right, so my Professor claims that it is impossible to find a vector field that has both div and curl zero and yet vanishes at infinity (that is, becomes the zero vector).</p>
<p>I don't know if it is correct. I decide to trust my professor and check for myself.<br>
So I conclude that,<br> if the vecto... | https://math.stackexchange.com/questions/4428280/divergence-and-curl-both-zero-what-can-we-say-about-behavior-at-infinity |
Question: <p>Let $f: R \rightarrow R$, $y_0,y_1,y_2 \in R$. We define divided differences:
$$[y_0;f]=f(y_0),$$
$$[y_0,y_1;f]=\frac{f(y_1)-f(y_0)}{y_1-y_0},$$
$$[y_0,y_1,y_2;f]=\frac{[y_1,y_2;f]-[y_0,y_1;f]}{y_2-y_0}.$$</p>
<p>Assume that for each $x \in R$ and $\varepsilon>0$ there exists a $\delta>0$ such t... | https://math.stackexchange.com/questions/185848/divided-differences-and-differentiability |
Question: <p>By the set of linear functions I mean the functions of the form <span class="math-container">$$f(x)=\alpha x,$$</span> for <span class="math-container">$\alpha\in\mathbb{R}$</span>. We clearly have for any <span class="math-container">$x,y\in[0,1]$</span> <span class="math-container">$$|f(x)-f(y)|=|\alpha|... | https://math.stackexchange.com/questions/4438313/is-the-set-of-linear-functions-from-0-1-to-mathbbr-equicontinuous |
Question: <p>I am investigating a strictly decreasing sequence $(a_i)_{i=0}^\infty$ in $(0, 1)$, with $\lim_{i\to\infty}a_i=0$, such that there exist constants $K>1$ and $m\in\mathbb{N}$ such that $$\frac{a_{i-1}^m}{K} \leq a_i \leq K a_{i-1}^m$$ for all $i$. Even though $K>1$, is it of the right lines to conclud... | https://math.stackexchange.com/questions/151811/doubly-exponential-sequence-behaviour-from-inequality |
Question: <p>Decide whether the following statement is true or false, justify the conclusion.</p>
<p>If $f$ is defined on $\mathbb R$ and $f(K)$ is compact whenever $K$ is compact, then $f$ is continuous on $\mathbb R.$</p>
<p>Does this hold based on the Preservation of Compact Sets: Let $f : A → \mathbb R$ be contin... | https://math.stackexchange.com/questions/1518100/preservation-of-compact-sets |
Question: <p>For a compactly supported, continuously differentiable function <span class="math-container">$f$</span>, do we have <span class="math-container">$\frac{f(x+h)-f(x)}{h}\to f'(x)$</span> uniformly? (Euclidean space)</p>
<p>I think this is true, since compact supported-ness is pretty strong condition and this... | https://math.stackexchange.com/questions/4455256/for-a-compactly-supported-continuously-differentiable-function-f-do-we-have |
Question: <p>By a unit circle, I just mean <span class="math-container">$D = S^1$</span> i.e. a unit disk in <span class="math-container">$R^2$</span>.</p>
<p>My game is to use some arbitrary sequence <span class="math-container">$x_n = (a_n,b_n)$</span> such that <span class="math-container">$b_n = \sqrt{1-(a_n)^2}$</... | https://math.stackexchange.com/questions/4439601/prove-that-a-unit-circle-contains-its-limit-points-that-it-is-closed-under-its |
Question: <blockquote>
<p>Does the function $f(x)=\sqrt{x}\sin(1/x),x\in(0,1],f(0)=0,$ satisfy the uniform Lipschitz condition $|f(x)-f(y)|<M|x-y|^{1/2},M>0$?</p>
</blockquote>
<p>Any help is appreciated.
Thanks</p>
Answer: <p>no, take $x =\frac{1}{2n\pi +\frac{\pi}{2}}$ and $y =\frac{1}{2n\pi -\frac{\pi}{2}... | https://math.stackexchange.com/questions/86111/uniform-lipschitz-condition |
Question: <blockquote>
<p>Suppose <span class="math-container">$z = a + bi$</span>, <span class="math-container">$w = u + iv$</span>, and
<span class="math-container">$$a = \left(\frac{|w| + u}{2}\right)^{1/2}, b = \left(\frac{|w| - u}{2}\right)^{1/2}$$</span>
Prove that <span class="math-container">$z^2 =w$</sp... | https://math.stackexchange.com/questions/3499058/rudin-chapter-1-exercise-10-hang-ups |
Question: <p>Suppose I have a deterministic sequence $\{t_n\}$ that is uniformly distributed on $[0,1]$ (for example $t_n = \{ \pi n \}$, i.e the fractional part of $\pi n$) and a decreasing function $f : \mathbb{R} \rightarrow [0,1]$.</p>
<p>I think it is reasonable to expect that
$\#\{t_k < f(k) : k \leq n \} \ap... | https://math.stackexchange.com/questions/21156/the-expected-value-of-a-deterministic-sequence |
Question: <p>E.T. Jaynes writes in "Probability Theory: The Logic of Science" (Chapter 17.7, page 531):</p>
<blockquote>
<p>Let a function y = f(x) have its domain of existence in the unit square <span class="math-container">$0 \leq x, y \leq 1$</span>; we wish to compute the integral
<span class="math-contai... | https://math.stackexchange.com/questions/1513568/uniform-grid-estimation-for-an-integral-by-e-t-jaynes |
Question: <p>In Rudin's POMA, the 7.29 theorem says:</p>
<blockquote>
<p>Let <span class="math-container">$\mathcal{B}$</span> be the uniform closure of an algebra <span class="math-container">$\mathcal{A}$</span> of bounded functions. Then <span class="math-container">$\mathcal{B}$</span> is a uniformly closed algebra... | https://math.stackexchange.com/questions/1466885/rudins-theorem-7-29 |
Question: <p>How do I show with the definition of continuity that <span class="math-container">$f : \mathbb{R} \to\mathbb{ R}$</span> defined by <span class="math-container">$f(x)= x$</span> for all <span class="math-container">$x\in\mathbb{R}$</span> is continuous?</p>
<p>I understand that this function is continuous,... | https://math.stackexchange.com/questions/2062471/show-that-the-real-valued-identity-function-is-continuous |
Question: <blockquote>
<p><strong>Problem:</strong></p>
<p><span class="math-container">$$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$</span></p>
</blockquote>
<p><strong>My proof:</strong></p>
<p>Let <span class="math-container">$x \in A \cap (B \cup C)$</span>. Then <span class="math-container">$x \in A$</span> a... | https://math.stackexchange.com/questions/2256835/basic-set-theory-proof-verification |
Question: <blockquote>
<p>Let the Minkowski-metric on <span class="math-container">$\mathbb{R}^3$</span> be (for a p<span class="math-container">$\in \mathbb{R}^3, v,w\in T_p\mathbb{R}^3$</span>) <span class="math-container">$<v,w>_p^{Mink}=-v_1w_1+v_2w_2+v_3w_3$</span>.</p>
<p>Show that:</p>
<p>i) <span class="m... | https://math.stackexchange.com/questions/2286962/minkowski-metric-induct-riamannian-metric |
Question: <blockquote>
<p>Let <span class="math-container">$f: [0,1] \rightarrow \mathbb{R} $</span> a continuous function and <span class="math-container">$f > 0$</span>. Let <span class="math-container">$F(x) = \int_0^x f(t)dt $</span> and let <span class="math-container">$F(1)=A$</span>.</p>
<p><strong>1</strong>... | https://math.stackexchange.com/questions/2356269/show-that-f-int-0x-ftdt-is-bijective-function-from-0-1-to-0-a |
Question: <p>I've encountered one tiny problem - I have to find tangent plane equation of the function <span class="math-container">$\space f(x,y)=y^3 - \sqrt{1-x^2y^2} \space$</span> in the point where it intersects with <span class="math-container">$\space y$</span>-axis. So <span class="math-container">$\space f(0,... | https://math.stackexchange.com/questions/3725351/tangent-plane-equation-in-the-point-of-intersection-with-space-y-axis |
Question: <p>Prove that there exist infinitely many positive real numbers <span class="math-container">$r$</span> such that the equation
<span class="math-container">$2^x +3^y + 5^z = r$</span> has no solution <span class="math-container">$(x, y, z) \subseteq \mathbb{Q} \times \mathbb{Q} \times \mathbb{Q}$</span>.</p>
... | https://math.stackexchange.com/questions/704990/prove-that-the-equation-has-no-solution |
Question: <p>I read a question stating that if $z$ is complex, then $|z|\leq 1$ is a closed set. I think this is just saying that the unit disk is a closed set. Why is that so?</p>
Answer: <p>Look at the complement of the set $S = \{ z \in \mathbb{C}: |z| \leq 1 \}$. The complement is given by $S^c = \{ z \in \mathbb{... | https://math.stackexchange.com/questions/57169/why-is-the-unit-disk-closed |
Question: <p>Consider <span class="math-container">$G\in C^{1}(\mathbb{R})$</span> such that <span class="math-container">$G(0)=0$</span>. Then there exists a constant <span class="math-container">$C$</span> such that <span class="math-container">$|G(s)|\leq C|s|$</span> for all <span class="math-container">$s\in [-M,M... | https://math.stackexchange.com/questions/3250395/pointwise-upper-bound |
Question: <p>Let $f : X \to Y$ and $g: Y \to Z$ be two functions, such that $(g \circ f) : X \to Z$ is bijective</p>
<p>Prove or refute that $f$ is surjective.</p>
<p>$\\$ </p>
<p>For $f$ to be surjective, $\forall b \in Y, \exists a \in X : f(a) = b$</p>
<p>Because $(g \circ f)$ is bijective, the sets X and Z have... | https://math.stackexchange.com/questions/2348789/prove-or-refute-that-f-is-surjective-are-my-thoughts-correct |
Question: <p>I'm trying to solve an exercise which its conclusion seems to be the title of this post. The exercise is:</p>
<blockquote>
<ol>
<li>Show that the function $h:\Bbb R\to [0,1[$ given by
$$h(t)=\begin{cases}
e^{-1/t^2} &\text{if } t\neq 0\\
0 &\text{otherwise}
\end{cases}$$
is $C^\in... | https://math.stackexchange.com/questions/198748/c-infty-version-of-urysohn-lemma-in-bbb-rn |
Question: <p>We have: </p>
<p>$|a|<1<|b| \iff |a|+|b| < 1 + |b| < 2|b|$</p>
<p>$\iff (|a|+|b|)² < (1+|b|)² < 4|b|² $</p>
<p>$ \iff|a|² + 2|a||b| + |b|² < 1 + 2|b| + |b|² < 4|b|²$</p>
<p>$ \iff 2|a||b| < 1 + 2|b| - |a|² < 3|b|² - |a|² $</p>
<p>I don't see how to proceed to get $ab \in ... | https://math.stackexchange.com/questions/2243342/a-b-in-mathbbr2-a-b-2-and-ab-show-that-1-in-a-b |
Question: <p>If $a < b$ then
$$((f \wedge b) -a)^-=(f-a)^-$$
where $f$ is a real valued function. I can't figure out why this holds. I know that $f^-=-(f\wedge 0)$ so I tried showing this from definitions. I would greatly appreciate any help.</p>
Answer: <p>It's easy checking cases…</p>
<p>For $f \ge b > a$ it... | https://math.stackexchange.com/questions/2124370/if-a-b-then-f-wedge-b-a-f-a |
Question: <p>Let $p\in\left[ 1,\infty\right) $ and $\left(
X_{0},\left\Vert \cdot\right\Vert _{p}\right) $ a normed linear space, where
$$
X_{0}=\left\{ \left( x_{n}\right) \in
\mathbb{R}
^{
\mathbb{N}
}\mid\exists n_{0}\in%
\mathbb{N}
,\forall n\in
\mathbb{N}
:n_{0}\geq n\Rightarrow x_{n}=0\right\} .
$$</p>
<p... | https://math.stackexchange.com/questions/2227604/properties-of-a-subspace-of-sequence |
Question: <p>An exercise from my book is as follows:</p>
<p>Assume that $x > 0$ for $x$ in $\mathbb{R}$ (Real numbers) then there is an $y$ in $\mathbb{N}$ (Natural Numbers) such that $1/y^3 < x$.</p>
<p>By the archimedean property, there exists an y in N such that $1/y < x$. How exactly would I continue on ... | https://math.stackexchange.com/questions/2125419/proof-using-archimedean-property |
Question: <p>I wonder if a matrix of second derivatives is positive definite and symmetric and the necessary condition of existance extremum is satisfy then exist exactly one extremum? </p>
Answer: <p>Maybe if this log likelihood function is a concave function
($\frac{\partial^2 ln L(\theta)}{\partial \theta_i \parti... | https://math.stackexchange.com/questions/2245651/existance-of-global-extremum |
Question: <p>how can I get $x$ from equation</p>
<p>$ e^{(\frac{-a}{x})}\frac{x}{a} - \frac{x}{a} = b - 1 $</p>
<p>I tried to play with logaritms but I had no luck. Can I even get $a$ from this equation?</p>
Answer: <p>The equation in unsolvable for $x$, the only way is a numerical approach</p>
| https://math.stackexchange.com/questions/2168267/get-x-from-exponential-aquation |
Question: <p>Let a$_n$ := $\frac {(n-1)(-1)^n}{n}$. Find lim sup a$_n$ and lim inf a$_n$.</p>
<p>I see that a$_n$ can either be $\frac {n-1}{n}$ when n is even or $\frac {-(n-1)}{n}$ when n is odd.</p>
<p>$\frac {n-1}{n}$ a$_2$ = $\frac {1}{2}$, a$_3$ = $\frac {2}{3}$, a$_4$ = $\frac {3}{4}$.</p>
<p>$\frac {-(n-1)}... | https://math.stackexchange.com/questions/2189040/how-to-find-lim-inf-lim-sup |
Question: <p>Let ${a_n}$ be a sequence and $b_n := \frac {2a_n + (-1)^n}{|a_n| +1}$ for $n$ in $\mathbb N$. Prove that ${b_n}$ has a convergent subsequence. </p>
<p>Attempt: </p>
<p>To show that the subsequence converges, I just have to show boundedness right? Is what I have here correct? Am I missing anything?</p>
... | https://math.stackexchange.com/questions/2189158/proving-convergent-subsequence |
Question: <p>Let $f(x) = x+\frac1x$. Show that $f(x)$ is uniformly continuous on $(0,\infty)$ for any fixed $c>0$?</p>
Answer: <p>Hint:</p>
<p>I suppose you mean $x\in[c,\infty)$
$$|\frac{1}{x}+x-\frac{1}{y}-y|=|1-\frac{1}{xy}||x-y|\leq (1+\frac{1}{xy})|x-y|\leq(1+\frac{1}{c^2})|x-y|$$</p>
<p>Hence $f(x)$ is Lips... | https://math.stackexchange.com/questions/1029490/uniform-continuity-of-fx-x-frac1x-on-0-infty-for-any-fixed-c0 |
Question: <p>Let $f:I \times J\rightarrow \mathbb R$ be a function of class $C^2$ in the open rectangle $I \times J \subset \mathbb R^2$. If $\frac{\partial^2f}{\partial x\partial y}\equiv 0$, prove that there exist $\phi:I \rightarrow \mathbb R$ and $\chi:J \rightarrow \mathbb R$ of class $C^2$ such that $f(x,y)=\phi(... | https://math.stackexchange.com/questions/1029606/decomposition-of-a-function-as-a-sum |
Question: <p>I am sucked by a very simple question. I want to find an element which minimizes the$\|A\|$(in $L^2$-norm ),where A is a random variable.
Then the textbook says the uniqueness will follow from strict convexity of the $L^2$ norm.</p>
<p>What does this mean? What should I do if I want to prove the uniquene... | https://math.stackexchange.com/questions/1029825/the-strict-convexity-of-the-l2-norm |
Question: <p>I have to prove that $\frac{e^x}{x^k} \to \infty$ for $x \to \infty$ with $k \in \mathbb N$</p>
<p>My idea is to calculate for $R \gt 0$ an $x_r$ so that for every $ x\gt x_r$ the inequation $\frac{e^x}{x^k} \gt R$ applies.
So, I struggle with solving the equation $\frac{e^{x_r}}{x_{r}^k} = R$ for $x_r$<... | https://math.stackexchange.com/questions/1026310/prove-divergence-of-a-series |
Question: <p>Here is a question I asked to myself. Does there exist a function $f(z)$ satisfying the following conditions:</p>
<ul>
<li>$f(z)$ is continuous on the strip $0 \leq \text{Re}(z) \leq 1$</li>
<li>$f(z)$ is <strong>holomorphic</strong> on the strip $0 < \text{Re}(z) < 1$</li>
<li>$f(a)=0$ where $a$ is... | https://math.stackexchange.com/questions/2234149/existence-of-a-function-satisfying-the-above-condition |
Question: <p>My professor used this term today, and I didn't get a chance to ask him what the heck it means. I can't find anything explicitly on the internet regarding the manner in which he used it.</p>
<p>for example, say we have a function $f(x)$ and a set $A$ where $f(x)$ is defined. If we say $f(x)$ is an extensi... | https://math.stackexchange.com/questions/2234524/the-phrase-extension-by-x |
Question: <p>Although this seems to be super basic, I cannot get my head around which steps I have to perform to transform $x^{2^{n+1}}\cdot x^{2^{n+1}}$ into $x^{2^{n+2}}$.</p>
<p>When I rewrite it as $x^{2^{{n+1}^2}}$ and the there are not braces, the power binds the strongest, that would lead to $x^{2^{n^2+2n+1}}$.... | https://math.stackexchange.com/questions/2192160/help-transforming-a-basic-equation |
Question: <p>I need to work out the Taylor series of $\frac{1}{1-x}$ in ascending powers of $(x-2)$ up to and including the term in $(x-2)^3$. </p>
<p>My method was to find all the derivatives (up to the third) and evaluate them at $x=0$ to give the standard geometric series formula</p>
<p>$\frac{1}{1-x} = 1 +x +x^2 ... | https://math.stackexchange.com/questions/2194096/taylor-series-for-frac11-x-in-ascending-powers-of-x-2 |
Question: <p>$\left( \mathsf{\mathbf{x}},\mathsf{\mathbf{y}}\right) $,
$_{[\mathsf{\mathbf{x}}=\left( x_{1},\cdots,x_{n}\right) \in R^{n}\text{
}\wedge\text{ }\mathsf{\mathbf{y}}=\left( y_{1},\cdots,y_{m}\right) \in
R^{m}]}$ $:=\left( x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}\right) \in R^{n+m}$</p>
<p>$W:=\left\{... | https://math.stackexchange.com/questions/2194909/is-this-proof-of-open-set-right |
Question: <p>On p. 21, Dieudonné states that: "... if $f$ is a continuous real function in a
bounded closed interval $I$, there is a smallest root and a largest root of the
equation $f(x)=0$ in $I$." </p>
<p>I haven't been able to convince myself of this basic
statement. I keep thinking of counterexamples that have on... | https://math.stackexchange.com/questions/2261139/question-about-statement-on-p-21-of-dieudonn%c3%a9s-infinitesimal-calculus |
Question: <blockquote>
<p>Let $m,n \ge 0$ be positive integers. Using induction on $n$ or otherwise, show that the improper integral $$\int_0^1 x^m \left(\log x\right)^n dx$$ exists, and give a closed-form expression for it.</p>
</blockquote>
<p>I'm a bit confused by the concept of induction on an integral, any help... | https://math.stackexchange.com/questions/2285577/use-induction-to-show-that-this-improper-integral-exists |
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