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

2013/06/28 by Tom McGaffey, McGaffey, Tom
Mathematics · #14J60 (Primary) 26A27 #26E35 (Secondary) #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #History and Theory of Mathematics #Mathematical and Theoretical Analysis #math.CA #math.DS #msc:14J60 #msc:26A27 #msc:26E35

paper · pdf · doi:10.48550/arxiv.1306.6900

arxiv created 2013/06/28 · openalex publication_date 2013/06/28 · arxiv updated 2013/07/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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 δ 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.

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