Nonstandard techniques and nowhere differentiable functions I: A dense family of generalized blancmange functions

TL;DR

This paper provides elementary nonstandard proofs demonstrating that generalized blancmange functions are nowhere differentiable and densely populate the space of continuous functions, with computational visualizations included.

Contribution

It introduces new nonstandard proof techniques for properties of generalized blancmange functions and explores their density among continuous functions.

Findings

Generalized blancmange functions are nowhere differentiable.

These functions are densely distributed among continuous functions.

Elementary nonstandard proofs simplify understanding of their properties.

Abstract

We will give an elementary nonstandard proof that the family of generalized blancmange functions are nowhere differentiable. The proof follows from the intuitive characterization of differentiability at a point as almost $\delta$ affine along with the transfer of the functional equations these functions satisfy. We also give elementary nonstandard proofs of the uniform density of these functions among continuous functions. Finally, we discuss work done with the Python programming language in displaying these functions.