Let X be a continuous random variable, where X is the amount of water in a bottle in milliliters. The probability that there is between $900$ milliliters and $1100$ milliliters of water in the bottle is given by the following equation:
$P(900 ≤ X ≤ 1100) = \int\limits_{900}^{1100}f(x)dx$,
where $f(x)$ is the probability density function. Thus, if we want to know the probability that there is exactly one liter of water in the bottle we look at the following equation:
$P(1000 ≤ X ≤ 1000) = \int\limits_{1000}^{1000}f(x)dx$
Obviously, $\int\limits_{1000}^{1000}f(x)dx = 0$. So the probability that X is exactly $1000$, $1100$ or any value you choose, is always equal to zero. Note that this is true for continuous random variables and not for discrete random variables.
Question: What's the general intuition behind $P(X = 1000) = 0$? I always thought that it had to do with the fact that the random variable was continuous; for every $\epsilon > 0$, I can find a value between $1000$ and $1000 + \epsilon$. This illustrates how unlikely it is to pick exactly $1000$, but I don't know if this correct.
Thanks in advance!
