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onesided derivatives

If the real function $f$ is defined in the point $x_{0}$ and on some interval left from this and if the lefthand onesided limit $\lim_{{h\to 0}}\frac{f(x_{0}+h)f(x_{0})}{h}$ exists, then this limit is defined to be the leftsided derivative of $f$ in $x_{0}$.

If the real function $f$ is defined in the point $x_{0}$ and on some interval right from this and if the righthand onesided limit $\lim_{{h\to 0+}}\frac{f(x_{0}+h)f(x_{0})}{h}$ exists, then this limit is defined to be the rightsided derivative of $f$ in $x_{0}$.
It’s apparent that if $f$ has both the leftsided and the rightsided derivative in the point $x_{0}$ and these are equal, then $f$ is differentiable in $x_{0}$ and $f^{{\prime}}(x_{0})$ equals to these onesided derivatives. Also inversely.
Example. The real function $x\mapsto x\sqrt{x}$ is defined for $x\geqq 0$ and differentiable for $x>0$ with $f^{{\prime}}(x)\equiv\frac{3}{2}\sqrt{x}$. The function also has the right derivative in $0$:
$\lim_{{h\to 0+}}\frac{h\sqrt{h}0\sqrt{0}}{h}=\lim_{{h\to 0+}}\sqrt{h}=0$ 
Remark. For a function $f\!:[a,\,b]\to\mathbb{R}$, to have a rightsided derivative at $x=a$ with value $d$, is equivalent to saying that there is an extension $g$ of $f$ to some open interval containing $[a,\,b]$ and satisfying $g^{{\prime}}(a)=d$. Similarly for leftsided derivatives.
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