$f(x)$ is a continuous function, for some fixed $a\in(0,1)$ and some real number A satisfy $$\lim\limits_{x\to0}\frac{f(x)-f(ax)}{x}=A$$ Prove that $f(x)$ is differentiable at $x=0$ and compute the value of $f'(0)$.
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1Welcome to MSE! Your question is closed because an almost identical problem has already been discussed in this community. This does not mean that your question has an issue, so I hope that this does not discourage you from asking other questions in this community. – Sangchul Lee Nov 21 '20 at 01:32