Convexity of the product of an Integer function and a convex function

Question

Assume $f_i (X)$ is a function that returns either $-1$ or $1$, and $g_i (X)$ is a (twice continuously-differentiable) convex function, and $h(X) = \sum_{i=1}^M f_i(X) \times g_i(X)$. $M$ is finite, and $X=\{ x_1,\dots,x_{N}\}$ is a finite set of variables with $N$ elements. My questions are:

1. Is h(X) continuous? (I guess not, because $f_i(X)$ is not continuous).
2. Is h(X) convex?
3. Does h(X) has a global (unique) minima?