Let f: R →R be a differentiable function such that f'(x) is continuous. Assume that f satsifies for any x, h e R: h f (x+ h) — f(х) %3D hf' (x + 2 (1) f(a+x)-f(a-x) defined everywhere on R (a) Prove...

c)
Prove that there exist m, l ∈ R such that for g defined in Part (b) (satisfying Equation

(2)) we have:

Extracted text: Let f: R →R be a differentiable function such that f'(x) is continuous. Assume that f satsifies for any x, h e R: h f (x+ h) — f(х) %3D hf' (x + 2 (1) f(a+x)-f(a-x) defined everywhere on R (a) Prove that for every a e R the function r(x) := 2х except zero, is constant. (b) Using Part (a) and Equation (1), prove that the function g(x) = f'(x) satisfies for any а, х € R: 8 (а + х) + g(а - х) = g(a). (2) 2 (c) Prove that there exist m, l e R such that for g defined in Part (b) (satisfying Equation (2)) we have: + l, for any integers p and n > 0. 2n = m • 2n Hint. Define l = g(0), and m = g(1) – g(0). (d) Using Part (c) and continuity of g, prove that g(x) = mx + l for some m, l e R. (e) Using Equation (1) and Part (d), prove that there exist a, b, c e R such that f(x) = ax? + bx + c.