What's the value of $\lim_{h\to0} P(a\leq X \leq a+h) $?

Question

The book that I am currently reading states that $$ \lim_{h\to0} P(a\leq X \leq a+h) = F(a+0)-F(a)=0 $$

I can't wrap my head around it. According to my understanding, shouldn't it be that $P(a\leq X \leq a) = P(X=a)$?

Edit: The X is a Continuous Random Variable.

Answer 1

As we are talking about limits, it makes sense to me to assume that $X$ a continuous variable (please provide that information)

The probability of a continuous variable taking any particular value is always $0$, so you are right, the limit is indeed $P(X=a) = 0$

Answer 2

You are correct to be weary. Writing $P(a\leq X \leq a+h) = P(X\leq a+h) - P(X\leq a) = F(a+h) - F(a)$ your book is assuming (right-) continuity of $F$ when they let $h$ tend to zero in the distribution function. Indeed $X$ may have some "mass" at $a$.