Abstract. In this paper we deal with regularity properties of functions $f$ and $g$ satisfying a functional inequality of the following type $$ |f(a(x,y))-f(a(x,z))|\le |g(b(x,y))-g(b(x,z))| ((x,y),(x,z) \in D), $$ where the real valued functions $a$ and $b$ defined on an open set $D\subset{\msbm R}^2$ enjoy certain sufficiently strong regularity properties. One of the main results states that if $g$ is pointwise Lipschitz on a dense subset of $b(D)$ (for instance if $g$ is differentiable on a dense subset) then $f$ is locally Lipschitz on $a(D)$. Another result says that if $f$ admits an inverse pointwise Lipschitz condition on a dense subset of $a(D)$ (for instance, if $f$ is differentiable on a dense subset with nonzero derivative), then $g$ is locally invertible with a locally Lipschitz inverse. The results so obtained have applications in the regularity theory of composite functional equations.

AMS Subject Classification (1991): 26D15, 26D07

Keyword(s): composite functional equation, regularity theory, local Lipschitz property, inverse local Lipschitz property

Received February 28, 2002. (Registered under 2918/2009.)