Have Object-Oriented Languages Missed a Trick with Class Function and its Subclasses?

Compared to functions in mathematics, functions in programming languages seem to be under classified. Functional programming languages based on the lambda calculus famously treat functions as first-class values. Object-oriented languages have adopted ``lambdas'', notably for call-back routines in event-based programming. Typically a programming language has functions, a function has a type, and some functions act on other functions and/or return functions but there is generally a lack of (i) ``class Function'' in the OO sense of the word class and particularly (ii) subclasses of Function for functions having specific properties. Some such classes are presented here and programmed in some popular programming languages as an experimental investigation into OO languages missing this opportunity.