Is this a parabola?

Consider the function $F(x) \; = \; \sum_{a=-100}^{100}|x+a| - 10100 \hspace{2cm} -100 \le x \le 100$ If $n$ is an integer between $-100$ and $100$ inclusive, then $\begin{aligned} F(n) & = \; \sum_{a=0}^{100+n}a + \sum_{a=0}^{100-n} a - 10100 \; = \; \tfrac12(100+n)(101+n) + \tfrac12(100-n)(101-n) - 10100 \; = \; n^2 \end{aligned}$ Since $F$ is linear between any pair of successive integers and since $x^2$ is a convex function, it follows that $F(x) > x^2$ for any noninteger $-100 \le x \le 100$ .

Thus the number of points where $f(x) = g(x)$ is the number of integers between $-100$ and $100$ that are common to the domains of both functions, namely the integers $n$ such that $-31 \le n \le 39$ . There are $\boxed{71}$ of these.