Let f be a real continuous function on [a, b], which is differentiable in (a, b) and such that there exists a positive real number L for which the following inequality holds

|f'(x)| L x (a, b).

Prove that for any two numbers x and y in [a, b] we have that |f(x) f(y)| L|x y|. (1)

Prove that the converse is true or give a counterexample.

Prove that if a real function f defined on (a, b) satisfies (1) for any two numbers x and y in (a, b) then f is uniformly continuous on (a, b).