Let

f : R--->R be differentiable with bounded derivative. Show that f is uniformly continuous.

(f(x) - f(y)) / (x - y) = f'(p)

for some p in (x,y) (by the mean value theorem) and if M > 0 is the bound

on all values of the derivative, so that

|f(x) - f(y) | / |x - y| <= M

or |f(x) - f(y)| <= M*|x-y| for all x,y.

Can you help please? Thanks