The only point at which we can be in doubt that h is continuous is at x=0. To prove that h is continuous at this point, you need to check that f(x)/x tends to f'(0) as x->0. Is this true?

To find out, consider how f'(0) is defined, i.e. f'(0) is the limit of some quotient, which might turn out to resemble f(x)/x.