Suppose $f:R->R$ is continuous, and that it has a continuous right derivative, i.e. the right-sided limit $$lim(\delta->0^+) (f(x+\delta)-f(x))/\delta$$
exists for all x $\in$ R and defines a continuous function on R.
Prove or find a counterexample: f must be C^1.
Thanks in advance,
Edit: I've tried the Mean Value Theorem (and trying to derive different upper bounds), which now I know is not valid to use. Also, moving some parts around in a difference quotient has not helped. So, I'm currently trying to use the fact that the continuous right derivative, call it g(x), is integrable, and apply the Fundamental Theorem of Calculus. Any help would be greatly appreciated. I will edit again to show my work in progress. Thanks,
Let $$F(x) = \int_a^x g(t)dt$$ where g(t) is the continuous right derivative, gotten from the original function, f.
Since g(x) is continuous, then by the Fundamental Theorem of Calculus, we have that $$F'(x) = g(x)$$ for all x $\in$ R.
I know I haven't said much yet. Not sure where to go from here...