A singularity is a point where things don’t behave the way they normally do, usually because some quantity goes to infinity (as many people have pointed out), but sometimes because something goes to zero.
For instance, a singularity of a vector field is a point where its value is zero. The reason is that [if the vector field is continuous and] you were to zoom in on a point where the value of the vector field is not zero, you’ll see essentially all vectors point in the same direction. But if you zoom in on a singularity, all sorts of weird things might happen (vectors all point to the singularity, or away from it, or they point to it in some parts and away from it in others, or the vectors circle around the singularity…).
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