Hi.
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It is a continuous function f on x_0? is it true or false? please prove or example
thanks
This could mean $\displaystyle f'_+(x_0)=\lim_{x\to x_0^+}f'(x)$ or $\displaystyle f'_+(x_0)=\lim_{x\to x_0^+}\frac{f(x)-f(x_0)}{x-x_0}$, which are two very different things. Regardless, though, you should make some effort in answering the question.
I realized that I myself got confused. No, if both one-sided derivatives exist, the function is not necessarily continuous. Both one-sided limits of the function exist (the proof of this is almost the same as for the fact that differentiable functions are continuous), but these limits are not necessarily the same. Consider an example shown here.