Residual pathologies

TL;DR

This paper demonstrates that certain pathological behaviors are typical in analysis, showing that they occur generically in various classes of mathematical objects, challenging the notion of these behaviors as mere exceptions.

Contribution

It provides three new examples where pathological behaviors are shown to be residual or generic in different analytical contexts.

Findings

Pathological behaviors are residual in classical analysis examples.

Derivative loss is generically observed in solutions to certain PDEs.

Counterexamples are typical, not exceptional, in the studied classes.

Abstract

Several counterexamples in analysis show the existence of some special object with some sort of pathological behavior. We present three different examples where the pathological behavior is not an isolated exception, but it is the "typical" behavior of the "generic" object in a suitable class, where here generic means residual in the sense of Baire category. The first example is the revisitation of a classical result concerning approximate differentiation. The second example is the derivative loss for solutions to linear wave equations with time-dependent Holder continuous propagation speed. The third result is the derivative loss for solutions to transport equations with non-Lipschitz velocity field.