a singular point
A singular point of a function is a point where the function fails to be holomorphic — where complex differentiability breaks down — while it remains holomorphic at points arbitrarily nearby. Loosely, it is a point where the smooth, well-behaved structure of f tears: a hole, a blow-up, or a place where f is simply not defined as a nice differentiable object.
For example 1/z is holomorphic everywhere except z = 0, which is a singular point (the function blows up there). The function tan z has singular points wherever cos z = 0, namely z = pi/2 + n pi. Log z has singular behavior along its branch cut and at the branch point 0. The simplest and most useful case is an ISOLATED singularity, where f is holomorphic on a small punctured disk around the point but not at the point itself.
Far from being defects to be avoided, singularities are where complex analysis becomes powerful. The whole machinery of Laurent series, residues, and contour integration is built to understand and exploit them; classifying an isolated singularity as removable, a pole, or essential controls how f behaves near it, and the residue theorem turns singularities into a tool for evaluating real integrals that have no elementary antiderivative.
f(z) = 1/(z^2 + 1) has singular points where z^2 + 1 = 0, namely z = i and z = -i; everywhere else it is holomorphic. These two isolated singularities are poles, and their residues evaluate integrals like the integral over the real line of 1/(x^2 + 1) dx = pi.
Singular points are not failures to fear but handles the theory grips — they power the residue calculus.
'Singular point' is broad; the rich classification (removable / pole / essential) applies to ISOLATED singularities. A branch point or a whole curve of bad behaviour is a different, non-isolated kind of singularity.