>Singularity defines the point in a function where it takes an infinite value,

It doesn't need to be infinite it can also be undefined or otherwise not well-behaved. For instance the function 1/x is never infinite for any finite value of x but it has a singularity because it is undefined at x=0. Another example is the piecewise function F(x)={x^2 if x>=0 {-x^2 if x<=0 for which x=0 has a definite value of y=0 but x=0 is considered a singularity for the purposes of calculus because it is not differentiable at that point.