We construct Lipschitz functions such that for all $s>0$ they are $s$-H\"older, and so proximally, subdifferentiable only on dyadic rationals and nowhere else. As applications we construct Lipschitz functions with prescribed H\"older and approximate subderivatives.

Keywords:

Lipschitz functions, Hölder subdifferential, proximal subdifferential, approximate subdifferential, symmetric subdifferential, Hölder smooth, dyadic rationals Lipschitz functions, Hölder subdifferential, proximal subdifferential, approximate subdifferential, symmetric subdifferential, Hölder smooth, dyadic rationals

MSC Classifications:

49J52, 26A16, 26A24 show english descriptions Nonsmooth analysis [See also 46G05, 58C50, 90C56]
Lipschitz (Holder) classes
Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 49J52 - Nonsmooth analysis [See also 46G05, 58C50, 90C56]
26A16 - Lipschitz (Holder) classes
26A24 - Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]

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