I need help with this problem in Real Analysis please, and please make it legible I often can't read the handwriting.

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). Leth: [a, c] - I be defined by h(x) = If(x), ifxela, 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