414/514 – Katsuura function

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.

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