First, the discrete Fourier transform(DFT) of a sequences $x=\{x_n\}_{n=1}^{N-1}$ is defined by $$ \mathcal{F}_k(x)= \sum_{n=0}^{N-1} x_ne^{-\frac{2\pi ikn}{N} } $$ for $k=0,1,..., N-1$. I have studied about Fourier transform, not DFT, which is defined by $$ \mathcal{F}(f)(\xi) = \hat{f}(\xi) = \int_{\mathbb{R}^n} f(x)e^{-2\pi i x\cdot\xi}\,dx, $$ (we suppose this Fourier transform is well-defined regardless of where $f$ is in any classes.)
My question is here. Roughly speaking, is DFT discrete version of Fourier transform? If it is, why is dividing by $N$ in $e^{-\frac{2\pi ikn}{N} }$ considered? I think that the $x_n, k, n$ and $\sum$ are correspond to $f, \xi,x$ and $\int$, respectively. But, what $N$ do? I need any help.