Vtyjkyuk

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.

Note: Below I'm assuming the usual definition of infinite derivatives, which is to say that $f'(0)=+infty$ if $lim_hto0(f(h)-f(0))/h=+infty$. 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 $f:[-1,1]toBbb R$ by
$$f(x)=begincases1-x,&(0<xle 1),
\0,&(x=0),
\-1-x,&(-1le x<0).endcases$$

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 $[a,b]$.

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.