  In this paper, we establish the Pexiderized stability of coboundaries and
cocycles and use them to investigate the Hyers--Ulam stability of some
functional equations. We prove that for each Banach algebra $A$, Banach
$A$-bimodule $X$ and positive integer $n, H^n(A,X)=0$ if and only if the $n$-th
cohomology group approximately vanishes.
