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.