\(
\newcommand{\cat}[1] {\mathrm{#1}}
\newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})}
\newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}}
\newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}}
\newcommand{\betaReduction}[0] {\rightarrow_{\beta}}
\newcommand{\betaEq}[0] {=_{\beta}}
\newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}}
\newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}}
\newcommand{\groupMul}[1] { \cdot_{\small{#1}}}
\newcommand{\groupAdd}[1] { +_{\small{#1}}}
\newcommand{\inv}[1] {#1^{-1} }
\newcommand{\bm}[1] { \boldsymbol{#1} }
\require{physics}
\require{ams}
\require{mathtools}
\)
Math and science::Analysis::Tao::09. Continuous functions on R
The intermediate value theorem
Continuous functions whose domain is closed enjoy two useful properties:
- the maximum principle
- the intermediate value theorem
This card covers the second.
Intermediate value theorem (my version)
Let \( a < b \) be reals, let \( X = [a, b] \), and let \( f: X \to \mathbb{R}
\) be a continuous function.
Then for every \( y \in [f_{min}, f_{max}] \), where
\( f_{min} \) and \( f_{max} \) are the minimum and maximum obtained by \( f \)
(which exist by the maximum principle), there is [...] such that
[...].
