 A map "formula_49" is said to be monotone if each of its fibers is connected i.e.for each element "formula_3" in formula_51 the (possibly empty) set formula_52 is connected.In functional analysis on a topological vector space formula_41, a (possibly non-linear) operator formula_54 is said to be a monotone operator if

Kachurovskii's theorem shows that convex functions on Banach spaces have monotonic operators as their derivatives.