CDF of a continuous and discrete random variable - attributes

Question

Something that bothers me as I was just doing an exercise.
We know that the CDF of a function must be:

1. Continuous from right side.
2. Monotonic up.
3. The limit at infinity is 1.
4. The limit at minus infinity is 0.

But if the random variable is discrete, then the CDF does not apply that? ( at least according to an exercise in a test ). Searched here for other topics of the same, could not find.
Is there a different definition for continuous random variable, discrete random variable and not discrete\continuous random variable, for the CDF?
If yes, please do tell me so, which attributes to which random variable it apply ( discrete, continuous, neither) and if there any extra attributes, I will be happy to hear

EDIT: Example of such:
$$F_X\left(t\right)=\begin{cases}0&t<\pi \\ \frac{1}{3}\:&\pi \le t<5\\ \frac{1}{2}\:&5\le t<10^6\\ \:1&t>10^6\end{cases}$$
And another one: $$F_X\left(t\right)=\begin{cases}0&t<1\\ 1-\left(1-\frac{1}{3}\right)^t&t\ge 1\end{cases}$$
And such a case of that it is not a CDF:
$$F_X\left(t\right)=\begin{cases}2^{-⌊t⌋-1}&t<0\\ 1-2^{-⌊t⌋-1}&t\ge 0\end{cases}$$ That is at my test, american question, first two possibly be CDF, the last one, is not CDF ( Of course all are for X discrete ).

Final Edit for post: I understood the solution, thanks for the comments!!