How to prove that a random variable is continuous? Prove that the random variable has a continuous distribution, or, $X$
“is continuous” for short, if and only if $P(X=x)=0$ for any $x$ in $R$.
Can anyone give some hint for this problem?
 A: The cdf. $F(c)= P[X \le c]$ is a non decreasing function. It is fairly straightforward to show that $F$ is always continuous from the right, that is,
$\lim_{t \downarrow x} F(t) = F(x)$.
There are various definitions of continuity for distributions.
The one you are using is that the distribution is continuous iff the cdf. $F$
is continuous as a function.
From the above remark, we see that the distribution is continuous iff the cdf. $F$
is continuous from the left.
Suppose $x_n \uparrow x$.
Noting that $\{x\} = \cap_n ((-\infty,x] \setminus (-\infty, x_n] ) $,
we see that $P\{x\} = F(x) - \lim_n F(x_n)$.
It follows that $F$ is continuous at $x$ iff $P\{x\} = 0$.
Another, different definition, is that the distribution is continuous iff
$F$ is absolutely continuous. For example, the cantor distribution is continuous but not absolutely continuous (cf. singular distributions).
A: HINT: note that
$$
\Pr[X\le c]-\Pr[X\le c-\delta]= F_X(c)-F_X(c-\delta)
$$
where $F_X$ is the distribution of $X$, and that $\Pr[X=c]=\Pr[X\le c]-\Pr[X<c]$.
