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In the question

Question on one-sided derivatives,

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 $\;\lim_{x\rightarrow {x_{0}}^{+}}(x-x_{0})f^{\prime}(x)\;$ ?

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  • $\begingroup$ Sorry. You are right. Let me correct the question. $\endgroup$ – Medo Jan 19 '19 at 15:06
  • $\begingroup$ Yes. This is a counterexample. Thanks a lot. $\endgroup$ – Medo Jan 19 '19 at 17:39
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Counterexample: $x_0=0$, $f(0)=0$, and $$f(x)=x^2\sin(x^{-3})$$ for $x\ne0$.

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