relation between DFT and continuous Fourier Transform

Greetings,

I have a function with a bounded support. For simplicity, let it be $\displaystyle \mathrm{supp} f(x) \subset (0,1)$. I would like to find $\displaystyle \hat f(z)$:

$\displaystyle \hat f(z) = \int_0^1 e^{-2 \pi i x z} f(x) dx $.

First, I thought that DFT defined by

$\displaystyle \hat F_k = \frac{1}{\sqrt{N}} \sum_{j=1}^N e^{-2 \pi i \frac{(j-1)(k-1)}{N}} F_j$

gave the following relation

if

$\displaystyle F_j = f\left(\frac{j-1}{N}\right)$

then

$\displaystyle \hat f(k-1) \approx \frac{\hat F_k}{\sqrt{N}}$

but, apparently, I was wrong.

So, what is the relation between Fourier transform and DFT?