### Proof lipschitz continuous

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##### Reference no: EM13131939

a) Let M>0, and let f:[a,b]-->R be a function which is continuous on [a,b] and differentiable on (a,b), and such that |f'(x)| <= M for all x belonging to (a,b) (derivative of f is bounded). Show that for any x,y belonging to [a,b] we have the inequality |f(x)-f(y)| <= M|x-y|. *apply mean value theorem.

b) Let f:R-->R be a differentiable function such that f' is bounded. Show that f is uniformly continuous. Use part a from above.

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