nowhere differentiable

nowhere differentiable

A

function

f

:

S

⊆

ℝ

→

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

∈

S

∩

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

f

⁢

(

x

)

=

∑

n

=

1

∞

b

n

⁢

cos

⁡

(

a

n

⁢

π

⁢

x

)

with

a

odd,

0

<

b

<

1

, and

a

⁢

b

>

1

+

3

2

⁢

π

.

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).

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 “” 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

]

.

Alternative examples of continuous and nowhere differentiable functions are given by the

sample paths

of standard

Brownian motion

, also known as the

Wiener process

or white noise. It was to handle such ill behaved processes, which have important applications to both pure and applied mathematics, that the techniques of stochastic calculus were developed.

Remark

. It can actually be shown that the Weierstrass function is in fact a fractal (that is, its graph in

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

⊆

ℝ

→

ℝ

n

is called nowhere differentiable if at each

x

∈

S

,

f

=

(

f

1

,

…

,

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

∈

S

. Given any function

f

:

S

⊆

ℝ

→

ℝ

n

, for any point

x

∈

S

, we say that

x

is of type

(

i

1

,

…

,

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

,

…

,

1

)

. Question:

is there a continuous nowhere differentiable function

f

:

S

⊆

R

→

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.

References

1

I. Stewart,

The Problems of Mathematics

,
Oxford University Press, 1987.

2

M. Yamaguti, M. Hata, J. Kigami,

Mathematics of Fractals

,
AMS, 1997

Title

nowhere differentiable

Canonical name

NowhereDifferentiable

Date of creation

2013-03-22 12:05:22

Last modified on

2013-03-22 12:05:22

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

24

Author

gel (22282)

Entry type

Definition

Classification

msc 26A15

Classification

msc 26A27

Related topic

Differentiable

Defines

Weierstrass function

Defines

nowhere differentiable curve

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