## 414/514 – Katsuura function

January 17, 2012

Here is Erron Kearns’s project from last term, on the Katsuura function, an example of a continuous nowhere differentiable function.

The presentation is nice: As usual with these functions, this one is defined as the limit of an iterative process, but the presentation makes it very clear the function is a uniform limit of continuous (piecewise linear) functions, and also provides us with a clear strategy to establish nowhere differentiability.

Actually, the function is presented in a similar spirit to many fractal constructions, where we start with a compact set $K$ and some continuous transformations $T_1,\dots,T_n$. This provides us with a sequence of compact sets, where we set $K_0=K$ and $K_{m+1}=\bigcup_{i=1}^n T_i(K_m)$. Under reasonable conditions, there are several natural ways of making sense of the limit of this sequence, which is again a compact set, call it $C$, and satisfies $C=\bigcup_{i=1}^n T_i(C)$, i.e., $C$ is a fixed point of a natural “continuous” operation on compact sets.

This same idea is used here, to define the Katsuura function, and its fractal-like properties can then be seen as the reason why it is nowhere differentiable.