Why isn't $x=1$ a discontinuity Problem:

Exercise 3.4.5. Let $\{X_n:n\in\Bbb N\}$ be a sequence of random variables such that for every $n\in\Bbb N$
$$\Bbb P\left(X=\frac1n\right)=\frac12,\qquad\Bbb P(X=1)=\frac12.$$
(a) Show that the sequence $\{X_n:n\in\Bbb N\}$ converges in distribution and find its limit.

Solution:

Exercise 3.4.5. (a) Let, for fixed $n\in\Bbb N$, $F_{X_n}(x)$ be the distribution function of $X_n$.  Then $$F_{X_n}(x)=\Bbb P(X_n\leq x)=\left\{\begin{array}{lc}0,&x<\frac1n, \\ \\\frac12,&\frac1n\leq x<1,\\\\ 1,& x\geq1,\end{array}\right.\quad\underset{n\to\infty}{\longrightarrow}\quad\begin{cases}0,&x\leq0,\\\frac12,&0<x<1,\\1,&x\geq1.\end{cases}$$
  Let the random variable $X$ be such that $$\Bbb P(X=0)=\frac12,\qquad\Bbb P(X=1)=\frac12.$$
  Then the distribution function $F_{X_n}(x)$ of $X$ is given by $$F_X(x)=\begin{cases}0,&x\leq0,\\\frac12,&0<x<1,\\1,&x\geq1.\end{cases}$$
  Thus, $F_{X_n}(x)\underset{n\to\infty}{\longrightarrow}F_X(x)$ for all $x\in\Bbb R$ except $x=0$ but $x=0$ is a discontinuity point of $F_X(x)$.  Hence, by the definition of convergence in distribution, $X_n\overset{d}{\longrightarrow}X.$

In the answer it is said that $x=0$ is a discontinuity. But why isn't $x=1$ a discontinuity? 
 A: The point $x=1$ is, as you said, a point of discontinuity.  The quoted text does not actually say it's a point of continuity.  All it's doing is checking that $F_{X_n}(x)\to F_X(x)$ at all points of continuity, that is, at all $x$ other than $0$ and $1$.  In fact $F_{X_n}(x)\to F_X(x)$ when $x=1$, too, even though this is not needed the convergence in distribution.
To show convergence is distribution all one needs to check is convergence at all the continuity points.  What happens at discontinuity points doesn't matter, one way or the other.  In your problem there are
two discontinuity points.  At one, $x=0$, the convergence fails, but that's allowed.  At the other, $x=1$, the convergence holds, which is also   allowed.
Imagine a school with required courses and optional courses, where you must pass the exams of the required courses, but are allowed to fail the exams of the optional courses.  The continuity points are like required courses; the discontinuity points are like optional courses.
Your example passes all the required exams, fails one optional exam, and passes the other optional exam. 
