Some properties of differentiable p-adic functions

TL;DR

This paper explores the algebraic structure of various subsets of p-adic differentiable functions, revealing their large and complex nature using lineability theory.

Contribution

It demonstrates that several classes of p-adic differentiable functions form large algebraic structures, highlighting their richness and diversity.

Findings

Sets of functions contain infinite dimensional algebraic structures

Differentiable functions with zero derivative are not Lipschitzian of any order

Functions differentiable on full measure sets may not be differentiable elsewhere

Abstract

In this paper, using the tools from the lineability theory, we distinguish certain subsets of $p$-adic differentiable functions. Specifically, we show that the following sets of functions are large enough to contain an infinite dimensional algebraic structure: (i) continuously differentiable but not strictly differentiable functions, (ii) strictly differentiable functions of order $r$ but not strictly differentiable of order $r+1$, (iii) strictly differentiable functions with zero derivative that are not Lipschitzian of any order $\alpha >1$, (iv) differentiable functions with unbounded derivative, and (v) continuous functions that are differentiable on a full set with respect to the Haar measure but not differentiable on its complement having cardinality the continuum.