Generalized Takagi-Van der Waerden Functions and their Subdifferentials

Abstract

<jats:p> We prove that for every countable dense subset D of a separable Hilbert space <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{H}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">H</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> there exists a continuous function with empty subdifferential at every point x outside D, meanwhile for every x in D the subdifferential at x is the whole space <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{H}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">H</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . </jats:p>