## 6.5. Differentiable Functions

### Examples 6.5.10(b):

If

Functions that satisfy the inequality
*f*is differentiable on*and***R***| f'(x) | M*for all*x*, then*| f(x) - f(y) | M | x - y |*for all numbers*x*,*y*.for some constant| f(x) - f(y) | M | x - y |

*M*are called

**Lipschitz**functions. Using those terms, we have to prove that if

*f*is differentiable and uniformly bounded then it is a Lipschitz function.

Take any two number *a*, *b*. By the mean value theorem we know that
there exists an *x* in *(a, b)* such that:

f'(x) =

Taking absolute values on both sides and moving the denominator to the other side we have

Since| f(b) - f(a) | = | f'(x) | | b - a |

*f'(x)*is uniformly bounded by

*M*, we therefore have

But that is exactly what we wanted to prove.| f(b) - f(a) | M | b - a |