= Singular defining maps can overestimate tangent spaces
{title2=$T_a(g^{-1}(0))\subseteq\ker Dg_a\text{ need not be equality}$}
If the zero set of a differentiable <function> $g$ is a <submanifold>, the <chain rule> always places its <tangent space> inside $\ker Dg_a$. 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 $Dg_0=0$ has kernel $\mathbb R$. One should therefore check the <constraint qualification> before using the kernel formula.
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