What is the purpose of
We know that in math mode one should write in natural language and not being so strict with math notation.
$f:A\to B$ is not "correct", the correct way would be
$f\colon A\to B$ (and variants).
I do not remember any math book that uses the colon as part of a sentence in math mode.
The only 2 uses I can give to
$:$ is when are learning about divisions (
$2:4=1:2$) and for scales (
Does (La)TeX give a meaning of
: in math mode?