According to this link: PDF
To translate the probability density $ρ(x)$ into a probability, imagine that $I_x$ is some small interval around the point $x$. Then, assuming $ρ$ is continuous, the probability that $X$ is in that interval will depend both on the density $ρ(x)$ and the length of the interval:
$Pr(X∈I_x)≈ρ(x) × $ Length of $I_x$
We don't have a true equality here, because the density $ρ$ may vary over the interval $I_x$. But the approximation becomes better and better as the interval $I_x$ shrinks around the point $x$, as $ρ$ will be come closer and closer to a constant inside that small interval.
My questions are:
Why is $Pr(X∈I_x)≈ρ(x) ×$ Length of $I_x$? Shouldn't it simply be $Pr(x∈A)=\int_{A}ρ(x)dx$?
Why as $I_x$ shrinks around the point $x$, will $ρ$ become closer and closer to a constant inside that small interval? Is it because as we reduce the interval, we zero in on $x$?
How and why the density $ρ$ may vary over the interval $I_x$?