# right derivative vs a right limit of the derivative times smallness factor

In the question

it is shown that if $$f$$ is differentiable on $$]x_0,x_0+\delta[$$ for some $$\delta>0$$, such that $$\;\lim_{x\rightarrow {x_{0}}^{+}}f^{\prime}(x)\;$$ exists, then $$f^{\prime}_{+}(x_{0})=\lim_{\epsilon\rightarrow 0^{+}} \frac{f(x_0+\epsilon)-f(x_0)}{\epsilon}$$ exists, but the converse is not true.

Question: Suppose $$f$$ is differentiable on $$]x_0,x_0+\delta[$$ for some $$\delta>0$$. Can $$f^{\prime}_{+}(x_{0})$$ exist but the limit $$\;\lim_{x\rightarrow {x_{0}}^{+}}(x-x_{0})f^{\prime}(x)\;$$ does not exist ?

In other words, does the existence of $$f^{\prime}_{+}(x_{0})$$ imply the existence of the limit $$\;\lim_{x\rightarrow {x_{0}}^{+}}(x-x_{0})f^{\prime}(x)\;$$ ?

• Sorry. You are right. Let me correct the question. – Medo Jan 19 '19 at 15:06
• Yes. This is a counterexample. Thanks a lot. – Medo Jan 19 '19 at 17:39

Counterexample: $$x_0=0$$, $$f(0)=0$$, and $$f(x)=x^2\sin(x^{-3})$$ for $$x\ne0$$.