a singular point

Run your finger along most curves and they feel smooth — at each point there is one clear direction the curve heads in, one tangent line. But some curves have bad spots: a sharp corner where the curve suddenly changes direction (a cusp), or a place where the curve crosses through itself (a node). At these spots the curve has no single well-defined tangent line. Such a point is called singular, and everywhere else the curve is smooth (also called regular or non-singular). Singular points are exactly where a curve misbehaves.

Here is the precise test, and it is concrete. Write the curve as f(x, y) = 0. Compute the two partial derivatives, f_x (treating y as constant) and f_y (treating x as constant). A point on the curve is SINGULAR when BOTH partial derivatives vanish there: f_x = 0 and f_y = 0 simultaneously, while f = 0. If at least one partial is non-zero, the implicit-function machinery hands you a unique tangent line and the point is smooth. When both vanish, the linear approximation collapses, the tangent direction is undetermined, and you get a node (two branches crossing, two tangent directions) or a cusp (a sharp spike, the two tangents fused) or worse. For y^2 = x^3, the origin is a cusp: f = y^2 - x^3, f_x = -3x^2 = 0 and f_y = 2y = 0 both hold at (0, 0).

Singular points are central, not a nuisance to be swept aside: a curve's deepest invariants (its genus, how it sits in space) depend on them, and 'resolving singularities' — smoothing them out by a controlled procedure — is one of the great themes of algebraic geometry, for which Heisuke Hironaka won the Fields Medal. A useful honest warning: singularity is a property of the EQUATION's geometry, not of how the picture looks at a glance. A curve can look perfectly smooth to the eye yet be singular in a way only the derivatives reveal, and conversely a self-crossing in a sloppy hand-drawing might just be two genuinely separate branches passing close by.