Singular defining maps can overestimate tangent spaces ($T_a(g^{-1}(0))\subseteq \ker D{g}_a\text{ need not be equality}$)

If the zero set of a differentiable function $g$ is a submanifold, the chain rule always places its tangent space inside $\ker D{g}_a$. Equality follows from the full-rank hypotheses of the regular level set theorem, but may fail for a singular defining function even when the zero set itself is smooth. For $g:\mathbb{R}\to \mathbb{R}$, $g(x)=x^2$, the zero set is the single point $\{0\}$. Its tangent space is zero, while $D{g}_0=0$ has kernel $\mathbb{R}$. One should therefore check the constraint qualification before using the kernel formula.