Editing Differentiable function (section)

===Semi-differentiability===
{{Main|Semi-differentiability}}
The above definition can be extended to define the derivative at [[Boundary (topology)|boundary points]]. The derivative of a function <math display="inline">f:A\to \mathbb{R}</math> defined on a closed subset <math display="inline">A\subsetneq \mathbb{R}</math> of the real numbers, evaluated at a boundary point <math display="inline">c</math>, can be defined as the following one-sided limit, where the argument <math display="inline">x</math> approaches <math display="inline">c</math> such that it is always within <math display="inline">A</math>:
:<math>f'(c)=\lim_{{\scriptstyle x\to c\atop\scriptstyle x\in A}}\frac{f(x)-f(c)}{x-c}.</math>

For <math display="inline">x</math> to remain within <math display="inline">A</math>, which is a subset of the reals, it follows that this limit will be defined as either
:<math>f'(c)=\lim_{x\to c^+}\frac{f(x)-f(c)}{x-c} \quad \text{or} \quad f'(c)=\lim_{x\to c^-}\frac{f(x)-f(c)}{x-c}.</math>