Symmetric and ordinary differentiation

Abstract

In 1927, A. Khintchine proved that a measurable symmetrically differentiable function <italic>f</italic> mapping the real line <italic>R</italic> into itself is differentiable in the ordinary sense at each point of <italic>R</italic> except possibly for a set of Lebesgue measure zero. Here it is shown that this exceptional set is also of the first Baire category; even more, it is shown to be a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ</mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-porous set of E. P. Dolženko.