Here is the question:
The lifetime $X$ of a machine has a continuous CDF. Find the conditional CDF and PDF given the event $C = \{X \gt t\} $.
Solution: $$F_X(x|X\gt t) = \mathbb{P}[X\le x| X\gt t] = \frac{\mathbb{P}[\{X\le x\} \cap \{X\gt t\}]}{\mathbb{P}[X\gt t]}$$
So we have:
$$ F_X(x|X\gt t) = \frac{F_X(x) - F_X(t)}{1 - F_X(t)} , \ x\gt t$$
According to my book, the PDF is given by: $$ f_X(x|X\gt t) = \frac{f_X(x)}{1 - F_X(t)} , \ x\ge t$$
The problem is $x=t$. Why $x=t$ is included in PDF? We should differentiate CDF with respect to $x$ and so $x=t$ shouldn't be included.
Density functions are not unique; they are only unique up to a set of measure $0$. Changing the value of a density function at a point makes no difference.