Intuition about theorem relating continuity and differentiability I am following Apostol's text on real analysis, and he has the following theorem:

What is the significance/motivation/intuition of this theorem? The proof seems circular/trivial by defining f* directly from equation (1). Why does it need to be a theorem that requires proof (i.e. why might it not hold)? What conclusions are we supposed to draw from it? 
(The text motivates it by saying that it allows us to reduce theorems on derivatives to theorems on continuity, but that is vague and unclear to me.) 
 A: If you draw the graph, then $f^*(x)$ is nothing but the slope of the line joining $(x,f(x))$ and $(c,f(c))$. So the intuition is that if this slope function varies continuously then $f$ is differentiable at $c$, i.e $f^*$ then has a well defined value at $c$.
Intuitively a function is not differentiable ar a point means in its graph that point is a sharp point, i.e left hand slope limit and right hand slope limit doesnot match.
A: The function $f^*$ is an approximation of the derivative $f'$ at $x_0$ in the sense that
$$f'(x_0)=\lim_{x\to x_0}\frac{f(x_0)-f(0)}{x_0-x}=\lim_{x\to x_0} f^*(x)$$
Which shows that $f^*$ function is continuous at $x=x_0$ iff $f$ is differentiable at $x_0$, since $f^*$ being continuous at $x_0$ means exactly the the limit on the RHS exists, which means exactly that $f$ is differentiable at $x_0$.
This shows how you can translate a problem about differentiability of $f$ into one about continuity of $f^*$, although I don't see how this would be helpful in practice.
