From what I could find, a singularity is a point at which an equation, surface, etc., blows up or becomes degenerate.
And a pole of a function is an isolated singular point a of single-valued character of an analytic function $f(z)$ of the complex variable $z$ for which $|f(z)|$ increases without bound when $z$ approaches $a$: $\lim_{z\rightarrow a}f(z) = \infty.$
I really don't fully understand this definition of a pole, like (what is an isolated singular point) and the limit says for $\lim_{z\rightarrow a}f(z) = \infty.$ What is $a$ that $z$ should approach?
$f = 1/(z-1) \ e^{z}$
Can I say $f$ has a singularity at $z = 1$ because we get $1/0$ at that point i.e. blows up and gives $\infty$?