I see what you mean. They give for x in (0,1], f(0)=0.
It's easy to check f'(x) is unbounded in this case. And then they conclude f'(x) is not integrable.....
It wouldnt be too hard to make such function with f(1)=1 and f(0)=0.
I'm confused right now. I think you may be right after all.
Anyway. To correct my solution in that case, I'll just 'steal' your idea. Again we assume
since we only need to show the existence of with we need to show that and for all is impossible. Then we're done.
Now, is increasing/decreasing. And since for all we can't have .
A contradiction.
Now again the intermediate value theorem guarantuees such that