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nowhere differentiable (Definition)

A function $ f:S\subseteq \mathbb{R}\to \mathbb{R}$ is said to be nowhere differentiable if it is not differentiable at any point in the domain $ S$ of $ f$.

It is easy to produce examples of nowhere differentiable functions. Indeed, any functions discontinuous at every point in $ S$ is nowhere differentiable, as seen in the example $ f(x)=x+1$ if $ x\in S\cap \mathbb{Q}$ and $ f(x)=x-1$ otherwise. Less trivial examples are those that are everywhere continuous, but nowhere differentiable (hence not an analytic function). These functions are simply termed continuous nowhere differentiable functions.

The prototypical example of a continuous nowhere differentiable function is the Weierstrass function. The formula for the Weierstrass function is

$\displaystyle f(x) = \sum_{n=1}^\infty b^n \cos(a^n \pi x) $

with $ a$ odd, $ 0 < b < 1$, and $ ab > 1 + \frac{3}{2}\pi$.

In the following series of figures, sequential renderings of the Weierstrass function are shown, each time zoomed in on the origin by a factor of ten. This shows the fractal nature of the function and illustrates how it is continuous, yet doesn't get any “smoother” as one looks closer (a prerequisite for differentiability).

\includegraphics[scale=0.75]{weier1}
\includegraphics[scale=0.75]{weier2}
\includegraphics[scale=0.75]{weier3}
Figure. Weierstrass function for $ a=9$, $ b=0.7$, with successive 10x zooms on the origin.

Notes

During eighteenth and early nineteenth centuries it was widely believed that every continuous function has a well defined tangent - at least at “almost all” points. As the Weierstrass function shows that this is clearly not the case. The function is named after Karl Weierstrass who presented it in a lecture for the Berlin Academy in 1872 [1].

Remark. It can actually be shown that the Weierstrass function is in fact a fractal (that is, its graph in $ \mathbb{R}^2$ has topological dimension $ 1$ but Hausdorff dimension $ >1$). Indeed, many other examples of continuous nowhere differentiable functions can be found in functions whose graphs are fractals. An example of a continuous nowhere differentiable function whose graph is not a fractal is the van der Waerden function.

Nowhere differentiability can be generalized. For example, a function $ f:S\subseteq \mathbb{R}\to \mathbb{R}^n$ is called nowhere differentiable if at each $ x\in S$, $ f=(f_1,\ldots,f_n)$ is not differentiable. This means that at least one of the coordinate functions $ f_i$ is not differentiable at $ x$, for every $ x\in S$. Given any function $ f:S\subseteq \mathbb{R}\to \mathbb{R}^n$, for any point $ x\in S$, we say that $ x$ is of type $ (i_1,\ldots,i_n)$ where $ i_j=1$ or 0 according to whether $ f_j$ is differentiable or not. If $ f$ is nowhere differentiable, then no points of $ S$ can be of type $ (1,\ldots,1)$. Question:

is there a continuous nowhere differentiable function $ f:S\subseteq \mathbb{R}\to \mathbb{R}^2$ such that for every pair of type-$ (0,1)$ points $ x_1<x_2$, there is a type-$ (1,0)$ point $ y$ such that $ x_1<y<x_2$, and every pair of type-$ (1,0)$ points $ y_1<y_2$, there is a type-$ (0,1)$ point $ x$ such that $ y_1<x<y_2$?
Remark. The above generalized definition leads to the definition of a nowhere differentiable curve: it is a curve such that it admits a nowhere differentiable parametrization. With this definition, we see that the graph of the Weierstrass function, as a curve, is nowhere differentiable. Other fractals, such as the Koch curve, is also nowhere differentiable.

Bibliography

1
I. Stewart, The Problems of Mathematics, Oxford University Press, 1987.
2
M. Yamaguti, M. Hata, J. Kigami, Mathematics of Fractals, AMS, 1997



"nowhere differentiable" is owned by CWoo. [ full author list (4) | owner history (1) ]
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See Also: differentiable

Also defines:  Weierstrass function, nowhere differentiable curve
Keywords:  differentiability, continuity
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Cross-references: Koch curve, curve, type, coordinate, van der Waerden function, Hausdorff dimension, dimension, graph, Karl Weierstrass, tangent, well defined, fractal, factor, origin, series, odd, analytic function, continuous, discontinuous, domain, point, differentiable, function
There are 4 references to this entry.

This is version 18 of nowhere differentiable, born on 2002-01-03, modified 2007-11-03.
Object id is 1181, canonical name is WeierstrassFunction.
Accessed 9510 times total.

Classification:
AMS MSC: 26A27 (Real functions :: Functions of one variable :: Nondifferentiability , discontinuous derivatives)
 26A15 (Real functions :: Functions of one variable :: Continuity and related questions )
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