Nowhere differentiable functions with respect to the position

2017/01/17 by Kavvadias, K., Makridis, K.
#26A27 #30H50 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1701.04875

Abstract

Let Ω be a bounded domain in ℂ such that ∂ Ω does not contain isolated points. Let R(Ω) be the space of uniform limits on Ω of rational functions with poles off Ω, endowed with the supremum norm. We prove that either generically all functions f in R(Ω) satisfy % \limsup_\substackz → z₀ z ∈ ∂ Ω | (f(z) - f(z₀))/(z - z₀) | = + ∞ for every z₀ ∈ ∂ Ω or no such function in R(Ω) meets this requirement. In the first case, the generic function f ∈ R(Ω) is nowhere differentiable on ∂ Ω with respect to the position. We give specific examples where each case of the previous dichotomy holds. We also extend the previous result to unbounded domains.

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