Let $X$ be a (EDIT) variety with a group structure. For $a\in A$, let $t_a$ be the translation on $X$: $t_a(x) = a+x$.
Why is the function $f:X\to \mathbf{C}$ given by $f(a) = \sum_{i} (-1)^i \mathrm{Tr}(t_a^\ast, H^i(X,\mathbf{C}))$ continuous?
Here I consider the usual singular cohomology with $\mathbf{C}$-coefficients. (The coefficients don't really matter. You can even take $\mathbf{Q}_{\ell}$-coefficients and work with $\ell$-adic cohomology.)
I call the function $f$ on $X$ the trace function. Note that one can use the Lefschetz trace formula to see that the image of $f$ lies in $\mathbf{Z}$.