Month: June 2022

Three basic theorems of calculus

Posted on by Nalin Pithwa

In the study of calculus, there are three basic theorems about continuous functions and on these theorems the rest of calculus depends. They are the following:

  • Intermediate Value Theorem: If f: [a,b ] \rightarrow \Re is continuous, and if r is a real number between f(a) and f(b), then there exists an element c \in [a,b] such that f(c)=r.
  • Maximum value theorem: If f: [a,b] \rightarrow \Re is continuous, then there exists an element c \in [a,b] such that f(x) \leq f(c) for every x \in [a,b].
  • Uniform continuity theorem: If f: [a,b] \rightarrow \Re is continuous, then given \epsilon >0 there exists a \delta>0 such that |f(x_{1})-f(x_{2})|<\epsilon for every pair of numbers x_{1}, x_{2} of [a,b] for which |x_{1}-x_{2}|<\delta.

These theorems are used in a number of places. The intermediate value theorem is used for instance in constructing inverse functions, such as \sqrt[3]{x} and \arcsin{x}, and the maximum value theorem is used for proving the mean value theorem for derivatives, upon which the two fundamental theorems of calculus depend. The uniform continuity theorem is used, among other things, for proving that every continuous function is integrable.

We have spoken of the these three theorems as theorems about continuous functions, But they can also be considered as theorems about the closed interval [a,b] of real numbers. The theorems depend not only on the continuity of f but also on properties of the topological space [a,b].

The property of the space [a,b] on which the intermediate value theorem depends is the property called connectedness, and the property on which the other two depend is the property called compactness.

As the three theorems quoted above are fundamental for the theory of calculus, so are the notions of connectedness and compactness fundamental in higher analysis, geometry and topology — indeed, in almost any subject for which the notion of topological space itself is relevant.

Reference: Topology by James Munkres, Second edition, Prentice Hall of India.

Regards,

Nalin Pithwa.

PS: I found above description to be another nice motivation to study topology! Of course, Hocking and Young succinctly mention that topology is an abstract study of the limit point concept. But Professor Munkres’s explanation is a bit more juicy !