The proof at the end starts off with: f(x) x->a = f(a) - the definition of continuous. So f(x) x->a - f(a)=0. Then simplifies to f'(a)*0, this however only holds if a function is continuous. If a function was not continuous then it would simplify to f'(a)*some constant. As f(x) x->a - f(a)=c - is not continuous.Which would make it not = 0. And in order to be differentiable x-a has to go to zero (by definition). That is how differentiable implies continuous.I hope this helps.####[]