Example: If f is differentiable on R and | f'(x) | M for all x, then | f(x) - f(y) | M | x - y | for all numbers x, y.

Functions that satisfy the inequality

• | f(x) - f(y) | M | x - y | for some constant 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

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

Since f'(x) is uniformly bounded by M, we therefore have

• | f(b) - f(a) | M | b - a |

But that is exactly what we wanted to prove.