A symmetric density property: monotonicity and the approximate symmetric derivative

Abstract:

The following is established: Let $W$ and $B$ be open sets of real numbers whose union has full measure. If for each $x$, the set $\{ h > 0|x - h \in W,x + h \in B\}$ has density zero at zero, then these sets are all empty. This is then used to prove the following: If $f$ is a continuous real valued function with a nonnegative lower approximate symmetric derivative, then $f$ is nondecreasing.