Mathematics

The key insight is recognizing that differentiability requires smoothness—no sharp corners or kinks. The absolute value function $f(x) = |x|$ has a sharp corner at $x = 0$ where the slope abruptly changes from $-1$ to $+1$. Since $f$ isn't even once differentiable there, it certainly can't be twice differentiable at that point. Everywhere else, $f$ is either $x$ or $-x$, both smooth polynomials with continuous derivatives.

The key insight is recognizing that differentiability requires smoothness—no sharp corners or kinks. The absolute value function $f(x) = |x|$ has a sharp corner at $x = 0$ where the slope abruptly changes from $-1$ to $+1$. Since $f$ isn't even once differentiable there, it certainly can't be twice differentiable at that point. Everywhere else, $f$ is either $x$ or $-x$, both smooth polynomials with continuous derivatives.