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Question Number 56706 by gunawan last updated on 22/Mar/19

$$f:[0,1)\to \mathbb{R}$$$$\underset{x\to 0}{\lim }f\left(x\right)=0$$$$\underset{x\to 0}{\lim }\frac{f\left(x\right)-f\left(\frac{x}{2}\right)}{x}=0$$$$\underset{x\to 0}{\lim }\frac{f\left(x\right)}{x}=...$$

Commented by Abdo msup. last updated on 22/Mar/19

$$\frac{x}{2}=t\Rightarrow {lim}_{x\to 0}\frac{f\left(x\right)-f\left(\frac{x}{2}\right)}{x}={lim}_{t\to 0}\frac{f\left(2t\right)-f\left(t\right)}{2t}=0$$$$\Rightarrow {lim}_{t\to 0}\frac{f\left(2t\right)}{2t}=\frac{1}{2}{lim}_{t\to 0}\frac{f\left(t\right)}{t}\Rightarrow l=\frac{l}{2}\Rightarrow l=0\Rightarrow$$$${lim}_{x\to 0}\frac{f\left(x\right)}{x}=0$$

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