For every continuous $f$ there is an absolutely continuous $g$ such that $[f=g]$ is not bilaterally strongly porous

Abstract:

For any Darboux function $f:\left [ {0,1} \right ] \to {\mathbf {R}}$ and any $0 < \delta < 1$ there is a point $x \in \left [ {0,1 - \delta } \right ]$ and a sequence ${x_n}$ such that (a) ${x_n} \in \left [ {x + {\delta ^{n + 1}},x + {\delta ^n}} \right ]\left ( {n = 1,2, \ldots } \right )$ and (b)$\sum \nolimits _{n = 2}^\infty {\left | {f\left ( {{x_n}} \right ) - f\left ( {{x_{n - 1}}} \right )} \right |} < + \infty$. Consequently, for every $f \in C\left [ {0,1} \right ]$ there is an absolutely continuous function $g$ such that $\left \{ {x:f\left ( x \right ) = g\left ( x \right )} \right \}$ is not bilaterally strongly porous.