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
is continuous, and if is a real number between and , then [...]. - Maximum value theorem
- If
is continuous, then [...]. - Uniform continuity theorem
- If
is continuous, then for every [...].
Applications in Calculus
- The intermediate value theorem is used for constructing inverse
functions, such as
and . - 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
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.