Math and science::Topology

# 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.