Question

Let α ∈ R. Prove that a function f : R → R is differentiable at α if and only if there is a function g : R → R which is continuous at α and satisfies f(x) - f(α) = g(x) (x - α.) for all x ∈ R.

Answer 1

Conversely, suppose f is differentiable at α, then

Hence, f(x) is differentiable at x = α, if  g(x) is continuous at x = α.