Weierstrass non-differentiable function

Question

In our analysis book it says that the following function proposed by Weierstrass is continuous everywhere, but nowhere differentiable.

$$f(x)=\sum _{n=1}^{∞}\frac{1}{2^n}\cos \left(3^nx\right)$$

The series must converge because it is the product of a convergent series and an alternating series, but the exponent inside cosine makes it tricky to evaluate.

When I graph it, it looks very strange indeed.

graph of function (n=1 to 100)

Would this be considered fractal? Is it differentiable analytically?

Answer 1

Your function is a Weierstrass function, which are of the form

$$W(x)=\sum_{k=0}^\infty a^k\cos(b^n\pi x)$$

Your function is of this form with $a=\tfrac12$ and $b=3$, since then $W(\frac x\pi)=f(x)$. Weierstrass functions are nowhere differentiable yet continuous, and so is your $f$. A quote from wikipedia:

Like fractals, the function exhibits self-similarity: every zoom is similar to the global plot.

So yes, it would be considered a fractal.

Read more about Weierstrass functions here.