Georg Cantor is famous for the first set theory (in "naive" terms) and the diagonal argument.
However Cantor is also credited with the Cantor Set and for constructing a continuous function which has no derivative almost everywhere.
How can one construct such a function?
Moreover how can one reason in order to come up with such a construction?
Thank you.
You are looking for the Weierstrass function. The function is a Fourier series with higher and higher frequencies. The terms are $a^n \cos (b^n \pi x)$ Since $a \lt 1$ the sum converges. If we take a term-by-term derivative we get $-(ab)^n \sin (b^n \pi x)$, so if $b$ is large enough the sum will diverge. It is continuous and nowhere differentiable.