We define and investigate the symmetric derivative for functions defined on a subset of the real line. We give an example of a continuous function with a positive symmetric derivative everywhere which is not monotonic. When the domain is measurable or has the Baire property, then a positive symmetric derivative does imply monotonicity on a big set.
"Symmetric derivatives on subsets of the real line and monotonicity.." Real Anal. Exchange 29 (2) 799 - 804, 2003-2004.