On the almost monotone convergence of sequences of continuous functions

A sequence (f (n) ) (n) of functions f (n) : X -> a"e almost decreases (increases) to a function f: X -> a"e if it pointwise converges to f and for each point x a X there is a positive integer n(x) such that f (n+1)(x) a parts per thousand currency sign f (n) (x) (f (n+1)(x) a parts per thousand yen f (n) (x)) for n a parts per thousand yen n(x). In this article I investigate this convergence in some families of continuous functions.