  We introduce a fairly general concept of functional equation for $k$-tuples
of functions $f_1,\dots,f_k\colon X \to Y$ between arbitrary sets. The
homomorphy equations for mappings between groups and other algebraic systems,
as well as various types of functional equations and recursion formulas
occurring in mathematical analysis or combinatorics, respectively, become
special cases (of systems) of such equations. Assuming that $X$ is a locally
compact and $Y$ is a completely regular topological space, we show that systems
of such functional equations, with parameters satisfying rather a modest
continuity condition, are stable in the following intuitive sense: Every
$k$-tuple of ``sufficiently continuous,'' ``reasonably bounded'' functions $X
\to Y$ satisfying the given system with a ``sufficient precision'' on a ``big
enough'' compact set is already ``arbitrarily close'' on an ``arbitrarily big''
compact set to a $k$-tuple of continuous functions solving the system. The
result is derived as a consequence of certain intuitively appealing
``almost-near'' principle using the relation of infinitesimal nearness
formulated in terms of nonstandard analysis.
