Editing Differentiable function (section)

==Differentiability in complex analysis==
{{main|Holomorphic function}}
In [[complex analysis]], complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing [[complex number]]s. So, a function <math display="inline">f:\mathbb{C}\to\mathbb{C}</math> is said to be differentiable at <math display="inline">x=a</math> when

:<math>f'(a)=\lim_{\underset{h\in\mathbb C}{h\to 0}}\frac{f(a+h)-f(a)}{h}.</math>

Although this definition looks similar to the differentiability of single-variable real functions, it is however a more restrictive condition. A function <math display="inline">f:\mathbb{C}\to\mathbb{C}</math>, that is complex-differentiable at a point <math display="inline">x=a</math> is automatically differentiable at that point, when viewed as a function <math>f:\mathbb{R}^2\to\mathbb{R}^2</math>. This is because the complex-differentiability implies that

:<math>\lim_{\underset{h\in\mathbb C}{h\to 0}}\frac{|f(a+h)-f(a)-f'(a)h|}{|h|}=0.</math>

However, a function <math display="inline">f:\mathbb{C}\to\mathbb{C}</math> can be differentiable as a multi-variable function, while not being complex-differentiable. For example, <math>f(z)=\frac{z+\overline{z}}{2}</math> is differentiable at every point, viewed as the 2-variable [[Real-valued function|real function]] <math>f(x,y)=x</math>, but it is not complex-differentiable at any point because the limit <math display="inline">\lim_{h\to 0}\frac{h+\bar h}{2h}</math> gives different values for different approaches to 0.

Any function that is complex-differentiable in a neighborhood of a point is called [[holomorphic function|holomorphic]] at that point. Such a function is necessarily infinitely differentiable, and in fact [[Analytic function|analytic]].