Is this a typo or a delicate point to understand? Suppose $X$ is a discrete random variable with distribution $F(t) = P(X \leq t)$. We know $P(X < t) = \lim_{s \to t^-} F(t) $. In trying to find a probability like $P(1/2 < X < 3/2) $, I would do 
$$ P(X<3/2) - P(X<1/2) = \lim_{x \to 3/2^-} F(t) - \lim_{x \to 1/2^- } F(t) $$
However, professor wrote 
$$ P(1/2 < X < 3/2) = P(X<3/2)  -  P(X \leq 1/2)  = P(X<3/2) - F(1/2) $$
is this a typo? or I am misunderstanding the concept?
 A: Your professor's answer is correct 
Remember that you are dealing with a discrete random variable, so you can't just assume that the probability of being exactly $1/2$ is zero, so you need to remember to subtract it off.
Your solution finds $P(1/2 \le X \lt 3/2)$ which isnt necessarily the same for discrete random variables
A: Not a typo. 
The notation $(1/2 < X < 3/2)$ is just a convenient notation for the event (=set) $\{ X \in (1/2,3/2)\; \}$. In particular, since the interval $(1/2,3/2)$ can be expressed as $A\setminus B$, where $A=(-\infty, 3/2)$ and $B=(-\infty, 1/2]$ (note the square bracket at the end), we have 
$$P(1/2 < X < 3/2) = P(A\setminus B) = P(X<3/2) - P(X\leqslant 1/2)$$ 
By definition, the last term is $P(X\leqslant 1/2)=F(1/2)$.

In other words, being strictly between $1/2$ and $3/2$ (that is, $1/2<X<3/2$) means being smaller than $3/2$ (that is, $X<3/2$) but not smaller than or equal to $1/2$ (that is, $X \leqslant 1/2$).
If $X=1/2$, then $X$ does not satisfy $1/2 < X < 3/2$.
