Differentiable

A Function is said to be differentiable at a point if its Derivative exists at that point. Let $z = x+iy$ and $f(z) = u(x,y)+iv(x,y)$ on some region $G$ containing the point $z_0$. If $f(z)$ satisfies the Cauchy-Riemann Equations and has continuous first Partial Derivatives at $z_0$, then $f'(z_0)$ exists and is given by

$$f'(z_0) = \lim_{z\to z_0} {f(z)-f(z_0)\over z-z_0},$$

and the function is said to be Complex Differentiable. Amazingly, there exist Continuous Functions which are nowhere differentiable. Two examples are the Blancmange Function and Weierstraß Function.

See also Blancmange Function, Cauchy-Riemann Equations, Complex Differentiable, Continuous Function, Derivative, Partial Derivative, Weierstraß Function