Continuous and Differentiable Functions

A function g is called continuous at the point t = t ₀ if the increment of g over small intervals is small,

If g is continuous at every point of its domain of definition, it is simply called continuous.

g is called differentiable at the point t = t ₀ if at that point

this constant C is denoted by g ?( t ₀). If g is differentiable at every point of its domain, it is called differentiable.

An important application of the derivative is a theorem on finite increments.

Theorem 1.1: (Mean Value Theorem)

If f is continuous on [a, b] and has a derivative on (a, b), then there is c, a < c < b, such that

Clearly, differentiability implies continuity, but not the other way around, as continuity states that the increment ? g converges to zero together with ? t, whereas differentiability states that this convergence is at the same rate or faster.

Example 1.1

The function g(t) = ?t is not differentiable at 0, as at this point

as t ? 0.

It is surprisingly difficult to construct an example of a continuous function which is not differentiable...