Let $f(x) = |x^2 - 1|$ for every real $x$. The figure shows the graph of $f$: a parabola whose part between $x = -1$ and $x = 1$ has been reflected above the $x$-axis, so that $f$ has corners at $x = -1$ and $x = 1$ and a local maximum at $x = 0$. Now let $h(x) = f(f(x))$, the function $f$ applied twice. A student argues that $h$ must keep the two corners of $f$, and must also gain new corners wherever $f(x) = 1$. At how many real numbers $x$ is $h$ not differentiable?

Fig. 1: The graph of y = |x^2 - 1| from x = -2.2 to x = 2.2: a parabola whose part between x = -1 and x = 1 has been reflected above the x-axis. The graph touches the x-axis at x = -1 and x = 1, where it has sharp corners, and has a rounded peak of height 1 at x = 0. A dashed horizontal line marks the level y = 1.