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 $0$:
$$\underset{h\to 0+}{lim}\frac{h\sqrt{h}0\sqrt{0}}{h}=\underset{h\to 0+}{lim}\sqrt{h}=0$$ 
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.
Title  onesided derivatives 
Canonical name  OnesidedDerivatives 
Date of creation  20130322 15:39:00 
Last modified on  20130322 15:39:00 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  9 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 26B05 
Classification  msc 26A24 
Synonym  left derivative 
Synonym  right derivative 
Related topic  Differentiable 
Related topic  OneSidedLimit 
Related topic  DifferntiableFunction 
Related topic  OneSidedContinuity 
Related topic  SemicubicalParabola 
Defines  leftsided derivative 
Defines  rightsided derivative 