The discrete Fourier transform (DFT) of $\{x_0,\ldots,x_{N-1}\}\subset\mathbb C$ is defined on Wikipedia as $$ X_k\stackrel{def}{=}\sum_{n=0}^{N-1}x_n\cdot e^{-2\pi ikn/N},\quad k\in\mathbb Z. $$ However, sometimes the DFT is defined for all real frequencies $\omega\in\mathbb R$ as $$ X_\omega\stackrel{def}{=}\sum_{n=0}^{N-1}x_n\cdot e^{-i\omega n},\quad \omega\in\mathbb R. $$ I am trying to understand why the frequencies \begin{equation} \tag{*}\label{freq} \frac{2\pi k}N \end{equation} with $k\in\mathbb Z$ are important. Are frequencies \eqref{freq} more important than other frequencies? Is it sufficient to consider the DFT only for frequencies $\eqref{freq}$? It seems that these frequencies are sometimes called the canonical frequencies. Is this a commonly used term or are there other terms for these frequencies?
Any help is much appreciated!
One reason to consider these frequencies is that the matrix $$\frac{1}{\sqrt{N}}\left[e^{-2\pi ikn/N}\right]_{k,n=0}^{N-1}$$ is unitary. This is nice for computation, since the inverse is just the conjugate transpose (not that you should ever form this matrix while computing the DFT). Also, it is clear that since the set of signals $\{(x_{0},\ldots,x_{N-1}):x_{i}\in\mathbb{C},0\leq i\leq N-1\}$ is a vector space, the rows/columns of this matrix form a Fourier basis for the space, corresponding to the complex exponential functions with frequencies $2\pi k/N$ (or $2\pi n/N,$ if you pick columns), which makes interpreting the $X_{k}$ easier (they are just the coefficients in the representation of the signal $x$ in terms of this basis).