Does value of differentiation as infinity implies continuity? I come across Mean value theorem proof which is attached below
Which has an assumption that f has derivative (finite or infinite ) at each interior point and continuity is assumed at endpoint.
In proof assumption used that function f is continuous over whole interval.
But I am not convinced with fact that f has infinite derivative and continuous at that point .
I tried to use definition I get $|f(x)-f(y)|=|x-y|f'(c)$ one side is infinite how to show for continuity.

Also Is it possible to have function which every where derivation infinity?
I think question is wrong As I could not imagine function every where like a verticle line. But Asking for in case exist?
Any help will be appreciated
Edit:


Sorry Everyone As In book already specified definition which already assume continuity of function to define the derivative.
 A: Note: Below I'm assuming the usual definition of infinite derivatives, which is to say that $f'(0)=+\infty$ if $\lim_{h\to0}(f(h)-f(0))/h=+\infty$. I've been told that that's not the definition Apostol uses...
In the usual statement of Rolle's theorem we assume that $f$ is differentiable on $(a,b)$, which is to say it has a finite derivative.
You're right, the existence of an infinite derivative at a point does not imply continuity. And in fact a simple counterexample for that is also a counterexample to the theorem as  stated: Define $f:[-1,1]\to\Bbb R$ by
$$f(x)=\begin{cases}1-x,&(0<x\le 1),
\\0,&(x=0),
\\-1-x,&(-1\le x<0).\end{cases}$$
Then $f$ satisfies all the hypotheses but there is no $x$ with $f'(x)=0$.
And of course it's easy to see where the proof fails: $f$ does not assume a max or min in $(-1,1)$.
What book did you find this nonsense in?
Note the theorem, with the usual definition of the derivative as above, becomes correct if we assume that $f$ is continuous on $[a,b]$.
A: It seems that Apostol requires continuity at the points where the derivative becomes infinity. At all the other points, the derivative is a real number, and hence the function is continuous there. It follows that your function is continuous everywhere.
