Can you help me with this Real Analysis problem, please?(Can you have the answer written too, please? I feel like I can read it easier that way. Thank you very much!)

Exercise 4.3.7. [Used in Lemma 7.3.4 and Theorem 7.3.12.] Let [a, b] C R and [b, c] C IR be non-degenerate closed bounded intervals, and let f: [a, b] - R and g: [b, c] - R be functions. Suppose that f(b) = g(b). Let h: [a, c] - R be defined by h(x) = (f(x), ifxe [a,b] g(x), ifxe [b,c]. (1) Suppose that f and g are differentiable, and that f'(b) = g'(b), where f'(b) and g' (b) are one-sided derivatives. Prove that h is differentiable