  If $f(x,y)$ is a real function satisfying $y>0$ and
$\sum_{r=0}^{n-1}f(x+ry,ny)=f(x,y)$ for $n=1,2,3,\ldots$, we say that $f(x,y)$
is an invariant function. Many special functions including Bernoulli
polynomials, Gamma function and Hurwitz zeta function are related to invariant
functions. In this paper we systematically investigate the properties of
invariant functions.
