Discrete Time Fourier Transform is given by

$X(ω)=∑_{n=-∞}^∞x(n).e^{-jωn}$ ………(1)

We know that DFT is given by

$X(k)=∑_{n=0}^{N-1}x(n).e^{\frac{-j 2πkn}{N}}$ ……….(2)

By comparing equation (1) and (2) we get

$X(k)=X(ω)|_{ω=\frac{2πk}{N}}$

This equations are equal if

i) Infinite summation is replaced by finite

ii) Continuous frequency variable is replaced by finite number of frequencies located at 2πk/NTs