For example, if we have $N$ data points, then they can be represented as a linear combination of complex exponentials of frequency in the range : 0 to $N-1$. How do we decide this range? First, this range consists of $N$ frequencies just like our $N$ data points, so I guess it'd be overkill if we used a bigger range, say, $0$ to $2N$. But, would any other set containing $N$ frequencies work fine too? Would the range $N+1$ to $2N$ work?
Again, in the continuous case, the range of frequencies used is $-\infty$ to $\infty$. How do we know that this range is sufficient to give back the original continuous functions?