Absolute value at a simple zero ($(|g|{)}_{\pm }^{\prime }(a)=\pm |g^{\prime }(a)|$)

Suppose a real function $g$ is differentiable at $a$, with $g(a)=0$ and $g^{\prime }(a)\ne 0$. The derivative definition gives $g(a+h)=g^{\prime }(a)h+o(|h|)$. The reverse triangle inequality then impliesThus $|g|$ has unequal one-sided derivatives, $|g^{\prime }(a)|$ on the right and $-|g^{\prime }(a)|$ on the left. More generally, multiplication by a function $w$ continuous at $a$ with $w(a)\ne 0$ gives right and left slopes $w(a)|g^{\prime }(a)|$ and $-w(a)|g^{\prime }(a)|$. This elementary expansion explains how a smooth oscillation can acquire infinitely many corners under the absolute value function, even after multiplication by a factor that makes it differentiable at an accumulation point.