On approximately convex functions

A realvalued function f defined on a real interval I is called (epsilon,delta)-convex if it satisfies f(tx+(1-t)y) less than or equal to tf(x)+(1-t)f(y) + epsilont (1-t)\x-y\ + delta for x, y is an element of I, t is an element of [0, 1]. The main results of the paper offer various characterizations for (epsilon,delta)-convexity. One of the main results states that f is (epsilon,delta)-convex for some positive epsilon and delta if and only if f can be decomposed into the sum of a convex function, a function with bounded supremum nor, and a function with bounded Lipschitz-modulus. In the special case epsilon=0, the results reduce to that of Hyers, Ulam, and Green obtained in 1952 concerning the so-called delta-convexity.