Functions not constant on fractal quasi-arcs of critical points

Norton, Alec

doi:10.1090/S0002-9939-1989-0969524-8

Functions not constant on fractal quasi-arcs of critical points
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Proc. Amer. Math. Soc. 106 (1989), 397-405 Request permission

Abstract:

This paper provides geometric sufficient conditions for an arc to be a critical set for some function not constant along that arc—an example of which was first discovered by Whitney in 1935. In particular, any fractal subarc of a quasi-circle has this property. The maximum degree of differentiability of the function is closely connected to the arc’s geometry.

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