This statement "the probability that a continuous variable exactly equals to a given value is zero" is commonly used.
is this an axiom, a property of continuous variable, a rule of integration or a conclusion of measure theory?
please provide a citation of a textbook or formally publication.
the wiki itself some answer linked is being tagged "unclear sources"
On
This washington handout gives
Property 1.1. If X is a continuous rrv, then For all $a\in\mathbb R, \ \mathbb P(X = a) = 0$
A random variable $X$ is by definition a continuous random variable if and only if $P(X=x)=0$ for every $x\in\mathbb R$.