I have a very very basic question on the CDF and limits

[Roni1985]

1. The problem statement, all variables and given/known data

1. For a random variable X, the function F defined by
F(x) = P(X <= x),−inf < x < inf
is called the cumulative distribution function of X. A property of every distribution function F is that
it is right continuous with left limits.
For the following functions, determine if they are right continuous and/or have left limits at the
indicated values. (A “yes” or “no” answer is insufficient. Justify your answers or no credit will be
given.)
(a) at x = 0 and x = 1 for G(x) = |x|/x .
(b) at x = 0 and x = 1 for H(x) = arctan(x).


2. Relevant equations

(a) at x = 0 and x = 1 for G(x) = |x|/x .
(b) at x = 0 and x = 1 for H(x) = arctan(x).

3. The attempt at a solution

Well, if we look at 'a', I was trying to find the limit when x->0+ and limit when x->0-. I am getting -1 and 1
but the function is not defined at x=0.
and it's not right continuous nor left continuous, and it has right and left limits, correct?

OR I don't get the question.

Can somebody tell me what I need to do here or lead me to the correct way ??

Thanks in advance,
Roni.

[Dick]

That sounds fine to me. Except I would say lim 0+ of |x|/x is 1. It's the limit as x->0 from the positive direction, right? OR I don't get the question. Or I misunderstand your notation.

[Roni1985]

That sounds fine to me. Except I would say lim 0+ of |x|/x is 1. It's the limit as x->0 from the positive direction, right? OR I don't get the question. Or I misunderstand your notation.

yes, that's what I meant, from the positive side it's 1 and from the negative side it's -1.

So, this equation is not a CDF ? as far as I understand, a CDF is right continuous and has left limits . But this one is neither right nor left continuous and has left and right limits, right ?

now what about arctan(x) ?

As far as I know, arctan is continuous from -inf to inf. So, it's right continuous but doesn't have left limit, right ? So it's not a CDF either?
If so, why did the professor mention the CDF properties ? O_o

[Dick]

Uh, don't know. Maybe the question isn't really about CDF's but about concepts of continuity and limits and mentioning CDFs was just a motivation. As far as I know, a good CDF f(x) has the property that lim x->-infinity is 0 and lim x->infinity is 1. Neither of those functions is good in that respect.

[Roni1985]

Uh, don't know. Maybe the question isn't really about CDF's but about concepts of continuity and limits and mentioning CDFs was just a motivation. As far as I know, a good CDF f(x) has the property that lim x->-infinity is 0 and lim x->infinity is 1. Neither of those functions is good in that respect.

I see, I guess you are right .
Thank you very much for your help :)