I have had this question for a very long time now, which is:-
"Why do we have $p(x)\,dx$ to be defined as the probability of $x$ lying in the interval $[x, x +dx]$ and not the interval $[x - \frac{dx}{2}, x + \frac{dx}{2}]$?"
Why is the preferred direction of evaluation an increase from the given value? Why is it not symmetric about $x$? Is it because of how Riemann sums are evaluated?