\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \)

Connectedness, compactness and some fundamental theorems of calculus

The following three theorems in calculus, theorems about functions from and to the reals, have generalizations in topology.

Intermediate value theorem

If \( f : [a, b] \to \mathbb{R} \) is continuous, and if \( r \) is a real number between \( f(a) \) and \( f(b) \), then [...].

Maximum value theorem

If \( f : [a, b] \to \mathbb{R} \) is continuous, then [...].

Uniform continuity theorem

If \( f : [a, b] \to \mathbb{R} \) is continuous, then for every \( \varepsilon > 0 \) [...].

Applications in Calculus

• The intermediate value theorem is used for constructing inverse functions, such as \( \sqrt[3]{x} \) and \( \arcsin(x) \).
• The maximum value theorem is used to prove the mean value theorem for derivatives, which in turn is used to prove the two fundamental theorems of calculus.
• The uniform continuity theorem is used for proving that every continuous function is integrable.

What is the concept in question: functions vs sets?

The three theorems can be considered to be describing facts about continuous functions; but shifting one's focus, one can view them as describing the nature of [...].

As topological properties

The topological property of the space \( [a, b] \) on which the intermediate value theorem depends is the topological property called [...].

The property which the maximum value theorem and the uniform continuity theorem depend on is called [...].

Both of these properties are fundamental to areas beyond calculus; they are fundamental to almost any area which can be represented in topology.