Editing Differentiable function (section)

==Differentiability of real functions of one variable==

A function <math>f:U\to\mathbb{R}</math>, defined on an open set <math display="inline">U\subset\mathbb{R}</math>, is said to be ''differentiable'' at <math>a\in U</math> if the derivative
:<math>f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}</math>
exists. This implies that the function is [[continuous function|continuous]] at {{mvar|a}}.

This function {{mvar|f}} is said to be ''differentiable'' on {{mvar|U}} if it is differentiable at every point of {{mvar|U}}. In this case, the derivative of {{mvar|f}} is thus a function from {{mvar|U}} into <math>\mathbb R.</math> 

A continuous function is not necessarily differentiable, but a differentiable function is necessarily [[continuous function|continuous]] (at every point where it is differentiable) as is shown below (in the section [[Differentiable function#Differentiability and continuity|Differentiability and continuity]]). A function is said to be ''continuously differentiable'' if its derivative is also a continuous function; there exist functions that are differentiable but not continuously differentiable (an example is given in the section [[Differentiable function#Differentiability classes|Differentiability classes]]).
===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>