3. Let f: R → R be a differentiable function such that f'(x) is continuous. Assume that f satsifies for any x, h e R: f(x+ h) – f(x) = hf' (x+). (1) 2 (a) Prove that for every a e R the function r(x)...

Question 3 a - c please

Extracted text: 3. Let f: R → R be a differentiable function such that f'(x) is continuous. Assume that f satsifies for any x, h e R: f(x+ h) – f(x) = hf' (x+). (1) 2 (a) Prove that for every a e R the function r(x) : f (a+x)-f(a-x) defined everywhere on R 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(a + x) + g(a – x) = 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: = m : 2n P + l, for any integers p and n > 0. 2n Hint. Define l = g(0), and m = 8(1) – g(0). %3D (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, ce R such that f(x) = ах? + bx + с.