Assertion: function | /r/HasProperty | continuous
Facets:
value: in an interval
type: TEMPORAL
value: at x = c
type: OTHER
value: at the value
type: OTHER
value: in their whole domain
type: LOCATION
value: at a point
type: TEMPORAL
value: at is equal
type: MANNER
value: at p if
type: TEMPORAL
value: at c
type: TEMPORAL
value: roughly
type: DEGREE
value: at a.
type: TEMPORAL
value: except 2
type: OTHER
value: in topology
type: LOCATION
value: with respect to some topology
type: OTHER
Source sentences:
text: Some caution must be exhibited, however, as it is not uncommon for the term map to denote a function with some sort of unspoken regularity assumption, e.g., in point-set topology, where "map" sometimes refers to a function which is continuous with respect to some topology.
text: For instance, a "map" is a "continuous function" in topology, a "linear transformation" in linear algebra, etc.
text: For many commonly used real functions, the domain is the whole set of real numbers, and the function is continuous and differentiable at every point of the domain.
text: Some functions are continuous in their whole domain, and not differentiable at some points.
text: A function f is continuous at a point a {\displaystyle a} which is interior to its domain, if, for every positive real number ε, there is a positive real number φ such that | f ( x )
text: A function is continuous if it is continuous at every point of its domain.
text: A function is continuous if it is continuous at every point of its domain.
text: If a is in the interior of the domain, the limit exists if and only if the function is continuous at a.
text: In particular, the many definitions of continuity employ the limit: roughly, a function is continuous if all of its limits agree with the values of the function.
text: A function is continuous at a limit point p of and in its domain if and only if f(p) is the (or, in the general case, a) limit of f(x) as x tends to p.
text: A function ƒ is said to be continuous at c if it is both defined at c and its value at c equals the limit of f as x approaches c:
text: The function f is continuous at p if and only if the limit of f(x) as x approaches p exists and is equal to f(p).
text: We are about to see that that is the definition of a function being "continuous at the value c."
text: Upon borrowing the word "continuous" from geometry then (Definition 1), we will say that the function is continuous at x = c.
text: We say that a function f(x) that is defined at x = c is continuous at x = c if the limit of f(x) as x approaches c is equal to the value of f(x) at x = c.
text: And so for a function to be continuous at x = c, the limit must exist as x approaches c, that is, the left- and right-hand limits -- those numbers -- must be equal.
text: If a function is continuous at every value in an interval, then we say that the function is continuous in that interval.
text: If a function is continuous at every value in an interval, then we say that the function is continuous in that interval.
text: And if a function is continuous in any interval, then we simply call it a continuous function.
text: Calculus is essentially about functions that are continuous at every value in their domains.
text: The function nevertheless is defined at all other values of x, and it is continuous at all other values.
text: In this same way, we could show that the function is continuous at all values of x except x = 2.
text: For a function to be continuous at x = c, it must exist at x
text: b) Define the function there so that it will be continuous.