Q:

Why $P(a \leq X < b) = P(a < X \leq b) = P(a < X < b) = P(a \leq X \leq b)$?

Let $F_X$ denote cumulative distribution function and $f_X$ denote probability density function.
If $X$ is a continuous random variable, then the following holds true:
$$\begin{align*}P(a \leq X < b) = P(a < X \leq b) = P(a < X < b) = P(a \leq X \leq b) &= \int_{a}^{b} f_X(x)dx \\ &= F_X(b) - F_X(a)\end{align*}$$
How can I easily explain that the used inequality symbols $\{<, \leq, >, \geq \}$ don't matter? Don't they? 

A:

$P(X=a)=0$ for  any $a$. Note that $[a,b]\setminus [a,b)=\{b\}$ so $P(a\leq X \leq b) -P(a\leq X  <b)=P(X=b)=0$. Similar argument holds for other cases. 

