View source for Non-differentiable function

A function that does not have a [[Differential|differential]]. In the case of functions of one variable it is a function that does not have a finite derivative. For example, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n0670101.png" /> is not differentiable at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n0670102.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n0670103.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n0670104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n0670105.png" /> is not only non-differentiable at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n0670106.png" />, 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n0670107.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n0670108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n0670109.png" /> is an odd natural number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n06701010.png" />. A simpler example, based on the same idea, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n06701011.png" /> is replaced by a simpler periodic function — a polygonal line — was constructed by B.L. van der Waerden. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n06701012.png" /> be the function defined for real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n06701013.png" /> as the absolute value of the difference between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n06701014.png" /> and the nearest integer. This function is linear on every interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n06701015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n06701016.png" /> is an integer; it is continuous and periodic with period 1. Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n06701017.png" /></td> </tr></table>

then van der Waerden's function is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n06701018.png" /></td> </tr></table>

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.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n067010a.gif" />

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n06701019.png" /></td> </tr></table>

is continuous at all points of the plane and has partial derivatives everywhere but it is not differentiable at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n06701020.png" />.



====Comments====
S. Banach proved that  "most"  continuous functions are nowhere differentiable. Specifically, he showed that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n06701021.png" /> denotes the space of all continuous real-valued functions on the unit interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n06701022.png" />, equipped with the uniform metric (sup norm), then the set of members of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n06701023.png" /> that have a finite right-hand derivative at some point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n06701024.png" /> is of the first Baire category (cf. [[Baire classes|Baire classes]]) in the complete metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067010/n06701025.png" />. Proof of this fact and of the nowhere differentiability of Weierstrass' example cited above can be found in [[#References|[a1]]]. A proof that van der Waerden's example has the stated properties can be found in [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,   K.R. Stromberg,   "Real and abstract analysis" , Springer  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K.R. Stromberg,   "Introduction to classical real analysis" , Wadsworth  (1981)</TD></TR></table>