I have the cumulative distribution function
$F_X(x) = \begin{cases} 0 & x<0\\ x & 0 \le x \le 1\\ 1 & x > 1 \end{cases}$
How am I meant to find, for example, $P(X = 0.25)$, using this function? My first instinct is to use the derivative of the function, but that would only give me its probability density, not the actual probability of X taking a certain value. What is the way of going about this? The fact that the variable is continuous is throwing me off. Any help is appreciated. Thanks!
$P(X=0.25)=P(X\leq 0.25 )-P(X < 0.25) =F_X(0.25)-F_X(0.25^{-})=0.25-0.25=0$
For any continues random variable $P(X=a)=0$ since
$P(X=a)=P(X\in \{a \})=\int_{x \in \{a \}} dP=0$ since measure of $\{a \}$ is zero.