# Non-differentiable function

A function that does not have a differential. In the case of functions of one variable it is a function that does not have a finite derivative. For example, the function is not differentiable at , though it is differentiable at that point from the left and from the right (i.e. it has finite left and right derivatives at that point). The continuous function if and is not only non-differentiable at , it has neither left nor right (and neither finite nor infinite) derivatives at that point.

The first examples of functions continuous on the entire real line but having no finite derivative at any point were constructed by B. Bolzano in 1830 (published in 1930) and by K. Weierstrass in 1860 (published in 1872). Weierstrass' function is the sum of the series

where , is an odd natural number and . A simpler example, based on the same idea, in which is replaced by a simpler periodic function — a polygonal line — was constructed by B.L. van der Waerden. Let be the function defined for real as the absolute value of the difference between and the nearest integer. This function is linear on every interval , where is an integer; it is continuous and periodic with period 1. Let

then van der Waerden's function is defined by

This function is continuous on the entire real line but does not have a finite derivative at any point. The first three partial sums of the series are shown in the figure.

Figure: n067010a

For functions of more than one variable, differentiability at a point is not equivalent to the existence of the partial derivatives at the point; there are examples of non-differentiable functions that have partial derivatives. For example, the function

is continuous at all points of the plane and has partial derivatives everywhere but it is not differentiable at .