\(
\newcommand{\cat}[1] {\mathrm{#1}}
\newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})}
\newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}}
\newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}}
\newcommand{\betaReduction}[0] {\rightarrow_{\beta}}
\newcommand{\betaEq}[0] {=_{\beta}}
\newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}}
\newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}}
\newcommand{\groupMul}[1] { \cdot_{\small{#1}}}
\newcommand{\inv}[1] {#1^{-1} }
\newcommand{\bm}[1] { \boldsymbol{#1} }
\require{physics}
\require{ams}
\)
Math and science::Analysis::Tao::09. Continuous functions on R
Equivalent formulations of function continuity
Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \)
be a function, let \( x_0 \) be an element of \( X \). Then the following
four statements are logically equivalent:
- \( f \) is continuous at \( x_0 \).
- [...]
- For any real \( \varepsilon > 0 \) there exists a real \( \delta > 0 \) such
that \( |f(x) - f(x_0)| < \varepsilon \) for all \(x \in X \) and
\( |x - x_0| < \delta \)