For example, the function f(x) = 1/x has a singularity at x = 0, where it explodes to ±∞ and isn't defined. The function g(x) = x (see absolute value) also has a singularity at x = 0, since it isn't differentiable there. The algebraic set defined by y^{2} = x^{2} in the (x,y) coordinate system has a singularity at (0,0) because it doesn't admit a tangent there. The algebraic set defined by y^{2} = x also has a singularity at (0,0), this time because it has a "corner" at that point.
In complex analysis, we distinguish three types of singularities. Suppose U is an open subset of C, a is an element of U and f is a holomorphic function defined on U{a}.
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