Given that you're talking about a discrete random variable over the integers, you should certainly know how the pmf behaves between those values - the probability of it taking any value in any interval strictly between (say) $1$ and $2$ is $0$.
Consequently, you do also know how the cdf $F(x)$ behaves, since it's just the sum of all the probabilities up to $x$. It doesn't matter how many zeroes you want add in, they don't change anything.
For further discussion of this point, see Wikipedia's Cumulative Distribution Function; Definition
... and the two sections immediately under that (Properties and Examples). You may find the drawing of a discrete cdf at the right hand side of the Properties section helpful (it's the top one).
Here's an example for a slightly different distribution than the one in your question (though it's broadly similar).