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We establish several examples of n-to-1 maps $f: \mathbold R \to \mathbold R$ satisfying the property that for every $y\in \mathbold R$ there exist exactly $n$ real numbers $x_1, x_2, ... , x_n$ in the domain such that $f(x_i) = y, i = 1, 2, ..., n.$ For odd $n$ those examples can be found to be continuous (and differentiable), whereas for even $n$ such examples are necessarily discontinuous. In connection with these type of functions, various invariant functions are considered, including even, odd, derivative invariant, stacked-periodic and $\omega-$invariant.

n-to-1 functions; stacked-periodic functions, even and odd functions