Some theorems whose \(\sigma\)-porous exceptional sets are not \(\sigma\)- symmetrically porous (Q1194679)

The author deals with assertions of the form ``A function \(f:\mathbb{R}\to\mathbb{R}\) has some property \(\mathcal P\) at every point except those in a \(\sigma\)-porous set''. It is known that \(\sigma\)-symmetrically porous sets are necessarily \(\sigma\)-porous and there are \(\sigma\)-porous sets which are not \(\sigma\)-symmetrically porous. There is a natural question wheather the above assertion can be formulated in a stronger form with exceptional sets to be \(\sigma\)-symmetrically porous. It is shown that in general the answer is negative, even if one is looking at fairly ``nice'' functions.