Continuous conditional probability Assume we have conditional density of continuous random variable $X_{1}$ with condition that $X_{2}=x_{2}^{0}$ where $x_{2}^{0}$ is some number. Then:
$$f(X_{1}|X_{2}=x_{2}^{0})=\frac{f(X_{1},X_{2}=x_{2}^{0})}{f_2(X_{2}=x_{2}^{0})}$$
My question concerns expression $f_2(X_{2}=x_{2}^{0})$ which equals $0$. 
In my textbook it is explained in such way that $X_{2}=x_{2}^{0}$ such that $x_{2}^{0} \epsilon\ B$ where $B$ is some subset of $R$ in which $f(x_{2})>0$ and $f(x_{2})=0$ for $x$ not in B. So my intuition is that $B$ is some infinitesimal interval around point $x_{2}^{0}$. But then, if $x_{2}^{0}$ is interval not a point, can we write $X_{2}=x_{2}^{0}$ ?
 A: Like others have already pointed out, the conundrum you are facing is because you are confusing probability with probability density.
Recall that we have an underlying probability space $(\Omega, \mathscr{F}, P)$, and $X_2:\Omega \to \mathbb{R}$. For some event $A \in \mathscr{F}$, we can express the integral relationship between probability density and probability itself as,
$$P(\{\omega \in \Omega : X_2(\omega) \in A\})\ \overset{\text{shorthand}}{=}\ P(X_2 \in A) = \int_A f_2(x_2)dx_2$$
For example, let the event be $A=[-10,12.4]$, then
$$P(X_2 \in [-10,12.4]) = \int_{-10}^{12.4} f_2(x_2)dx_2$$
If the event is $A=\{3\}$ then we might say,
$$P(X_2 \in \{3\}) = \int_{3}^{3} f_2(x_2)dx_2 = 0$$
but this does not imply that $f_2(x_2) \equiv 0$ or that $f_2(3)=0$.
It may ease confusion if you only put random variables (capital letters) as arguments of $P$ and only "samples" (really dummy variables, lowercase letters) as arguments of $f$. Saying $f_1(X_1)$ is weird, but $f_1(x_1)$ is not.
A: If you have continuous RVs $X_1,X_2$ with joint density function $ f_{X_1X_2}(x_1,x_2)$ then you can define the conditional joint density given $X_2 = x_2^0$ as $$ f_{X_1|X_2}(x_1|x_2^0) = \frac{f_{X_1X_2}(x_1,x_2^0)}{f_{X_2}(x_2^0)} $$ where $f_{X_2}(x_2)$ is the marginal density of $X_2,$ given by $\int_{-\infty}^\infty f_{X_1X_2}(x_1,x_2)dx_1.$
This way everything is in terms of densities. 
You are right that there's little sense writing $f_2(X_2=x_2^0).$ In general I'm not a fan of the notation you're using cause it confuses random variables with the variables corresponding to them in density functions. 
One notation I've seen that is like what your book is talking about is to write the joint density function as $P(X_1\in dx_1,X_2\in dx_2).$ In this notation $dx_1$ refers to an infinitessimal interval around $x_1$ and similarly for $x_2.$ This describes pretty well what a density function is.
