Polya's space-filling curve is a continuous function from the unit interval onto a solid right triangle in the plane. In 1973, P. Lax proved a remarkable theorem about this function, showing that, depending on the value of the smallest of the two acute angles of the triangle, there are three possibilities: it is either nowhere differentiable, differentiable almost nowhere with uncountably many exceptions, or differentiable almost everywhere with uncountably many exceptions.
In 2006, H. Okamoto introduced a one-parameter family of self-affine functions from the unit interval onto itself which includes Perkins' function, the Katsuura function and the classical ternary Cantor function. When investigating the differentiability of these functions he found a result remarkably similar to Lax's. In this talk I will show what ties these two, seemingly very different, functions together and explain why Lax's and Okamoto's results are two manifestations of the same general principle.