  We study the continuous solutions of several classical functional equations
by using the properties of the spaces of continuous functions which are
invariant under some elementary linear trans-formations. Concretely, we use
that the sets of continuous solutions of certain equations are closed vector
subspaces of $C(\mathbb{C}^d,\mathbb{C})$ which are invariant under affine
transformations $T_{a,b}(f)(z)=f(az+b)$, or closed vector subspaces of
$C(\mathbb{R}^d,\mathbb{R})$ which are translation and dilation invariant.
These spaces have been recently classified by Sternfeld and Weit, and Pinkus,
respectively, so that we use this information to give a direct characterization
of the continuous solutions of the corresponding functional equations.
