Regular and singular points of an analytic set

Remarks

Example: for the cusp $A = \set{z_1^2 = z_2^3}$ in $\C^2$, every point $a \ne 0$ is regular, since the gradient $(2 z_1, -3 z_2^2)$ of $f = z_1^2 - z_2^3$ is nonzero there and the holomorphic implicit function theorem applies. The origin is singular. If $A$ were a complex submanifold near $0$, it would have dimension $1$ and be a graph over its tangent line at $0$. The secant directions from $0$ to $(t^3, t^2)$ tend to the $z_2$-axis, so this tangent line is the $z_2$-axis, but the projection $(t^3, t^2) \mapsto t^2$ to it is two-to-one near $0$. Regularity is a property of the set, not of the equations. The set $A = \set{z_1^2 = 0}$ in $\C^2$ is the line $\set{z_1 = 0}$, so all its points are regular, although the gradient of $z_1^2$ vanishes on $A$. The set $\operatorname{reg} A$ is open in $A$ by definition, and it is dense in $A$. The set $\operatorname{sing} A$ is a closed analytic subset of $X$ that is nowhere dense in $A$; this is a theorem, not a direct consequence of the definition.