Qualitative symmetric differentiation (Q1308863)

The notions of qualitative limits, qualitative continuity and qualitative derivatives were introduced by S. Marcus. Let \(f: \mathbb{R}\to\mathbb{R}\) and \(x_ 0\in\mathbb{R}\). The qualitative limit superior from the right of \(f\) at \(x_ 0\) is defined as \(q\)-\(\lim\sup_{x\to x_{0^ +}} f(x)= \inf\{y:\{x: f(x)>y\}\) is of first category in a right neighborhood of \(x_ 0\}\). The qualitative limit inferior from the right of \(f\) at \(x_ 0\), \(q\)-\(\lim\inf_{x\to x_{0^ +}}f(x)\), is defined analogously. If \(q\)-\(\lim\sup_{x\to x_{0^ +}}f(x)=q\)-\(\lim\inf_{x\to x_{0^ +}}f(x)\), the common value is called the qualitative limit from the right of \(f\) at \(x_ 0\). Qualitative limits from the left and qualitative limits are then defined and denoted in the obvious fashion. A function \(f\) is said to be qualitatively continuous at \(x_ 0\) if \(q\)- \(\lim_{x\to x_ 0} f(x)= f(x_ 0)\). The qualitative derivative (symmetric derivative) of \(f\) at \(x\) is defined to be \[ f_ q'(x)=q\text{-}\lim_{h\to 0}(f(x+h)- f(x))/h\quad (f_{sq}'(x)= q\text{-}\lim_{h\to0}(f(x+ h)- f(x- h))/2h). \] The authors show that the set of points at which a function is qualitatively continuous and finitely qualitatively symmetrically differentiable but not qualitatively differentiable is a \(\sigma\)-symmetrically porous set. It is also shown that the above result holds if instead of using category as an indicator of size of a set the notion of Lebesgue measure is used.