Let \(D\subseteq\mathbb R\) (\(D\) is a subset of real numbers) and let \(f:D\to\mathbb R\) be a function. The function $f$ is called right-differentiable at a point $x\in D,$ if the limit. \[f'_+(x):=\lim_{\substack{\xi\searrow x\\\xi\in D\setminus\{x\}}}\frac {f(\xi)-f(x)}{\xi-x}\] at $x$ from above exists. In this case, $f'_+(x)$ is called the right-derivative of $f$ at $x$.
The function $f$ is called left-differentiable at a point $x\in D,$ if the limit. \[f'_-(x):=\lim_{\substack{\xi\nearrow x\\\xi\in D\setminus\{x\}}}\frac {f(\xi)-f(x)}{\xi-x}\] at $x$ from below exists. In this case, $f'_-(x)$ is called the left-derivative of $f$ at $x$.
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