A function f : A->R is given to be differentiable

A function f : A->R is given to be differentiable. A is an open interval.

Let a₁ and a₂ (a₁ < a₂) are two real no. in A s.t. f'(a₁ ) is not equal to f'(a₂).

If a₀ is any no. between f'(a₁) and f'(a₂) then

(i) prove that there exist some x₀ in (a₁ ,a₂ ) s.t. f(x₀)=a₀.

(iii) Can we conclude that derivative of f has to be continuous on interval A.

Note: This is complete question, no info missing..concepts which may be used to solve this question: Convergence of sequences, continuity of a function, differentiability of a function, Intermediate value theorem, considering cases to prove a thing.