=== Definitions 1 and 2 are equivalent === We reformulate Definition 1 into Definition 1' by: :substituting the definition of $\omega_f \left({U \cap X}\right)$ into the definition of $\omega_f \left({x}\right)$ Definition 1': :$\displaystyle \omega_f \left({x}\right) := \inf_{U \mathop \in \mathcal N_x} \left({\sup_{y, z \mathop \in U \cap X} \left\lvert{f \left({y}\right) - f \left({z}\right)}\right\rvert}\right)$ We reformulate Definition 2 into Definition 2' by: :substituting the definition of $\omega_f \left({\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X}\right)$ into the definition of $\omega_f \left({x}\right)$ :changing set condition into sup tag Definition 2': :$\displaystyle \omega_f \left({x}\right) := \inf_{\epsilon \mathop \in \R_{>0}} \left({\sup_{y, z \mathop \in \left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X} \left\lvert{f \left({y}\right) - f \left({z}\right)}\right\rvert}\right)$ The theorem text of Oscillation at Point (Infimum) equals Oscillation at Point (Epsilon-Neighborhood) also contains two definitions of Oscillation at a Point for a real function $f$. We call them definitions a and b. Definition a: Let $\mathcal N_x$ be the set of open subset neighborhoods of $x$. :$\omega_f \left({x}\right) = \displaystyle \inf \left\{{\omega_f \left({I}\right): I \in \mathcal N_x}\right\}$ where: :$\omega_f \left({I}\right) = \displaystyle \sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap X}\right\}$ Definition b: :$\omega^E_f \left({x}\right) = \inf \left\{{\omega_f \left({I}\right): I \in E_x}\right\}$ where: :$E_x$ is the set of $\epsilon$-neighborhoods of $x$ We reformulate Definition a into Definition a' by: :substituting the definition of $\omega_f \left({I}\right)$ into the definition of $\omega_f \left({x}\right)$ :changing set conditions into sup tags :renaming parameter set $I$ to $U$ Definition a': :$\displaystyle \omega_f \left({x}\right) = \inf_{U \mathop \in \mathcal N_x} \left({\sup_{y, z \mathop \in U \cap X} \left\lvert{f \left({y}\right) - f \left({z}\right)}\right\rvert}\right)$ We reformulate Definition b into Definition b' by: :substituting the definition of $\omega_f \left({I}\right)$ into the definition of $\omega^E_f \left({x}\right)$ :changing set conditions into sup tags :using $\epsilon$ as parameter instead of $I$ Definition b': :$\displaystyle \omega^E_f \left({x}\right) = \inf_{\epsilon \mathop \in \R_{>0}} \left({\sup_{y, z \mathop \in \left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X} \left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert}\right)$ We observe that definitions 1' and a' are the same, so they are equivalent. Therefore: :definitions 1 and a are equivalent Also, definitions 2' and b' are the same, so they are equivalent. Therefore: :definitions 2 and b are equivalent We have: :definitions 1 and a are equivalent :definitions a and b are equivalent by Oscillation at Point (Infimum) equals Oscillation at Point (Epsilon-Neighborhood) :definitions b and 2 are equivalent Therefore: :definitions 1 and 2 are equivalent {{qed|lemma}} === Definitions 1 and 3 are equivalent === We reformulate Definition 1 into Definition 1' by: :substituting the definition of $\omega_f \left({U \cap X}\right)$ into the definition of $\omega_f \left({x}\right)$ Definition 1': :$\displaystyle \omega_f \left({x}\right) := \inf_{U \mathop \in \mathcal N_x} \left({\sup_{y, z \mathop \in U \cap X} \left\lvert{f \left({y}\right) - f \left({z}\right)}\right\rvert}\right)$ We reformulate Definition 3 into Definition 3' by: :substituting the definition of $\omega_f \left({\left({x - h \,.\,.\, x + h}\right) \cap X}\right)$ into the definition of $\omega_f \left({x}\right)$ Definition 3': :$\displaystyle \omega_f \left({x}\right) := \lim_{h \mathop \to 0^+} \left({\sup_{y, z \mathop \in \left({x - h \,.\,.\, x + h}\right) \cap X} \left\lvert{f \left({y}\right) - f \left({z}\right)}\right\rvert}\right)$ The theorem text of Oscillation at Point (Infimum) equals Oscillation at Point (Limit) also contains two definitions of Oscillation at a Point for a real function $f$. We call them definitions a and b. Definition a: Let $\mathcal N_x$ be the set of open subset neighborhoods of $x$. :$\omega_f \left({x}\right) = \displaystyle \inf \left\{{\omega_f \left({I}\right): I \in \mathcal N_x}\right\}$ where: :$\omega_f \left({I}\right) = \displaystyle \sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap X}\right\}$ Definition b: :$\omega_f \left({x}\right) = \displaystyle \lim_{h \mathop \to 0^+} \omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right)$ We reformulate Definition a into Definition a' by: :substituting the definition of $\omega_f \left({I}\right)$ into the definition of $\omega_f \left({x}\right)$ :changing set conditions into sup tags :renaming parameter set $I$ to $U$ Definition a': :$\displaystyle \omega_f \left({x}\right) = \inf_{U \mathop \in \mathcal N_x} \left({\sup_{y, z \mathop \in U \cap X} \left\lvert{f \left({y}\right) - f \left({z}\right)}\right\rvert}\right)$ We reformulate Definition b into Definition b' by: :substituting the definition of $\omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right)$ (by using the definition of $\omega_f \left({I}\right)$) into the definition of $\omega_f \left({x}\right)$ :changing set condition into sup tag Definition b': :$\displaystyle \omega_f \left({x}\right) = \lim_{h \mathop \to 0^+} \left({\sup_{y, z \mathop \in \left({x - h \,.\,.\, x + h}\right) \cap X} \left\lvert{f \left({y}\right) - f \left({z}\right)}\right\rvert}\right)$ We observe that definitions 1' and a' are the same, so they are equivalent. Therefore: :definitions 1 and a are equivalent Also, definitions 3' and b' are the same, so they are equivalent. Therefore: :definitions 3 and b are equivalent We have: :definitions 1 and a are equivalent :definitions a and b are equivalent by Oscillation at Point (Infimum) equals Oscillation at Point (Limit) :definitions b and 3 are equivalent Therefore: :definitions 1 and 3 are equivalent {{qed|lemma}} === Definitions 2 and 3 are equivalent === From: :Definitions 2 and 1 are equivalent :Definitions 1 and 3 are equivalent follows: :Definitions 2 and 3 are equivalent {{qed|lemma}} Thus all definitions listed in the theorem text are equivalent. {{qed}} Category:Real Analysis 0lnoe98f2nec430ufqyzytt4ue8dcan