By: Jan Wuzyk

In this article I am going to show that nowhere differentiable functions do in fact exists and give a few examples, some of which are relatively modern. But first I’m going to try to answer a question that is, in my opinion, too rarely discussed in mathematics classes, ”Why do we care?”.

# Why we care

To answer this question we have to look into the history of mathematics. In 1821 Augustin-Louis Cauchy published his seminal book, Cours d’Analyse, this is generally recognized as the first serious attempt to put calculus on a rigours footing[Com] , mainly through introducing rigorous definitions of limits,continuity and differentiability among others, and the definitions that go with them2. This was also time the integral was defined as an area instead of simply as the antiderivative.

It should be noted that Cauchy by no means closed the issue of rigour in analysis but he provided a starting point. In fact much of his work was wrong or not actually rigorous but it inspired a slew of later mathematicians who turned it bit by bit into the model for rigour that is modern analysis. For example:

1841: Karl Weierstrass published ”Zur Theorie der Potenzreihen” first using uniform convergence of a series in a proof.

1853: Bernhard Riemann published ”ber die Darstellbarkeit einer Function durch eine trigonometrische Reihe” first defining the Riemman Integral.

Now in 1872 analysis was barely 50 years old and many theorems and definitions we now consider relatively boring and straightforward were new or not even discovered yet. It is in this setting that Weierstrass published an example of a continuous nowhere differential function. This contributed yet another example of a pathological function that most thought shouldn’t exist, and even more evidence that the then new rigour in analysis was entirely necessary.

Why shouldn’t such a function exist? Well here’s an argument someone might have made, intuitively a continuous functions is a function whose graph we can draw without lifting our pen from the paper. Now it’s easy to make such a function not differentiable at a point, simply make a corner at that point, but to make a corner we must have edges and those edges should be pretty flat and thus differentiable. Now matter how rough we make our function if we look close enough we should find the flattish edges of each corner.

Of course now that we are used to such aberrations this argument looks pretty flawed but it’s not impossible that when people still barely understood analysis this sort of thing might appear relatively convincing.

So that’s why nowhere differential functions matter, not because they themselves are all that interesting but because they show that our intuition can lie and that to do correct mathematics we must be careful and rigorous.

# Some Examples

Now that I have made such a fuss about a nowhere differentiable continuous function I should show you one. Firstly Weierstrass famous one (not the first but the most famous).

for 0 < b < 1 and a an integer > 1 such that ab > 1 + 3π/2. The bound on ab was eventually improved to ab > 1 but that is the original function. We can’t actually draw it since it, in a sense, has ”infinte detail” but here’s a picture after summing the first few terms, it should give some impression of what the complete beast would look like. Note that it is self similar when zoomed.

[Eey]

In fact it has a whole family of nowhere differentiable functions associated with it

which more directly shows that these are in fact special Fourier Series which skip lots of terms called lacunary Fourier series from which a few pathological functions come.

These aren’t the only nowhere differentiable functions, at the end of this article I prove that the function

where φ is defined as in the next section, is continuous but nowhere differentiable.

There is also the actual original nowhere differentiable continuous function, the Bolzano function which is defined geometrically, I wont go into this but if you’re interested see[Thi].

Riemann also came quite close to giving a nowhere differentiable function with

but this function is actually differentiable at points of the form pπ/q where p,q are odd as was show much later[Ger].

Space filling curves such as the Peano and Seirpinksi curves, which prove something quite interesting on their own, also happen to be nowhere differentiable.

Probably the most well know nowhere differentiable curves nowadays are fractals which computers have made much more accessible since they can draw pictures of them.

The final example I’ll give is Brownian motion which is used to model anything from the movement of particles to the fluctuations of stock prices.

So in spite of their initial oddness continuous nowhere differentiable do end up occurring naturally in mathematics and the rigorous foundations the early examples helped create have become crucial for modern mathematics. I’ll end with the fact that in a technical sense the vast majority of continuous functions actually are nowhere differentiable[Bow].

# Finally, a Theorem

Theorem 1. Define φ(x) on [−1,1] as |x| and then extend φ to R by setting φ(x + 2n) = φ(x) for n ∈ Z. Let

then f(x) is continuous but nowhere differentiable.

The proof of nowhere differentiability relies on constructing a sequence δm such that as δm the value of the derivative quotient, , blows up proving that the function can’t be differentiable at any x. The construction and proof of the blow up is technical so just bear with it, it ends up giving us the answer we need.

Proof. For continuity simply note that so the sum must converge uniformly by the Weierstrass M test. Since each partial sum is continuous this then implies f(x) is continuous.

Now for differentiability, fix x0 ∈ R. Pick such that no integer is between 4mx0 and 4m(x0 + δm). This is possible since the length of

)) = 1 so it can contain at most one integer and

this must be in either  or (4 )) so the other contains no integers.

Now consider

If n > m then 4 for some k ∈ Z so

φ(4n(x0 + δm)) − φ(4nx0) = φ(4nx0) − φ(4nx0) = 0 thus for n > m, γn = 0.

If n < m then

where the inequality follows from the fact that |φ(x) − φ(y)| ≤ |x y|.

Finally if n=m

The second equality follows since when x + δ and x are both between the same integers φ(x) is linear so φ(x + δ) = φ(x) + δ.

Now finally we consider the sequencewhich approaches x0 as m → ∞

and the the quotient.

thus as x0 + δm x0 dm → ∞. This implies lim doesn’t exist so the function can’t be differentiable at x0. Since x0 was arbitrary the function is nowhere differentiable.

# References

[Bow] Most           Continuous           Functions              are          Nowhere                Differentiable. http://homepages.math.uic.edu/ marker/math414/fs.pdf

[Com] Gower, Timothy (Hrsg.): Princeton Companion to Mathematics

[Eey] Eeyore22: WeierstrassFunction. https://commons.wikimedia.org/wiki/

File:WeierstrassFunction.svg

[Ger] Gerver, J.: The Differentiability of the Riemann Function at Certain Rational Multiples of pi.

[Thi]    Thim,         Johan:     Continuous           Nowhere                Differentiable      Functions. https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf

 It should be noted that there were many mathematicians working on this before Cauchy but Cauchy is the first seriously successful attempt at rigour 2Somewhat ironically part of the reason for this book was pedagogy.

 An actual definition Fourier gave

 How clear is this post?