Calculus/Differentiable functions

Introduction

In the study of calculus, we consider a special class of functions which are differentiable. We call them differentiable functions. Just as a continuous function is a function that is continuous at every point, so a differentiable function is one that is differentiable at every point.

We say that the function f is differentiable at the point x if the derivative f'(x) exists. Since the derivative f'(x) of f at x is defined as a limit, it's quite possible that it won't exist. For example, if f is not even continuous at x then it can't be differentiable there (exercise). Continuity of f at x is not enough, though.

For example, consider the function $f(x)=|x|$. The function f is differentiable at every point x other than x = 0. To see that it's differentiable at x for $x\ne 0$, you can either informally draw a graph and "see" it, or you can prove it with the epsilon-delta definition of differentiability as a limiting process. At x=0, though, the direction of the graph changes suddenly at that point, so there is no well-defined tangent line (and so no derivative) for f at 0. We call the point 0 a "critical point" of f.

For more information about differentiation refer to the differentiation section of this textbook.

Extreme value theorem

The extreme value theorem states that for function f(x), continuous on the closed interval [a,b] . f(x) must attain its maximum and minimum value each at least once. Mathematically, there exists numbers m and M such that

$m\le f(x)\le M$

And there exist some c and d such that

$f(c)=m$

and

$f(d)=M$

To formulate a proof of the extreme value theorem is quite hard, because it is so obviously true and that it almost seems a proof is unnecessary. However, various proofs are available. Refer to Extreme value theorem for details.

An important result that the extreme value theorem establishes is the following: Suppose that f is differentiable and that f has a local maximum or a local minimum at x = c. Then f'(c) = 0.