# Why are discrete-time Fourier series and discrete Fourier transform only defined on integer $k$?

In ordinary Fourier series/transform of a continuous signal $f(t)$, fourier frequencies $\omega$ of series/transforms can be any of $\mathbb{C}$, not just $\mathbb{Z}$.

But why is it the case that discrete-time Fourier series and transforms have $N$ frequenicies, defined by the number of samples?

For a function defined on a compact interval $[a,b]$ (or a periodic function), you have a correspondence with Fourier series, indexed by $k \in \mathbb Z$.
For a function defined on the entire line, you have a correspondence via the Fourier transform, here the transform variable $\xi$ is defined on the line as well.