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As you can see the above equation, DTFT is calculated from sample x[n] which is discrete sample of x(t).
But calculated X(w) is continuous, even though it is calculated from discrete value of x[n] as per equation 1.
How it is possible to get continuous output from discrete input??
What you may be missing is that $\omega$ is continuous. You can plug any value $\omega \in \mathbb{R}$ and get a number $X_{2\pi}(\omega)$. This is why $X_{2\pi}$ is continuous.
Discrete signals in one domain correspond to periodic signals in the other domain, and vice verse.
Continuous signals in one domain correspond to aperiodic signals in the other domain, and vice versa.
In this case, discrete and aperiodic time-domain signal $x[n]$ corresponds to a periodic and continuous frequency-domain signal $X(\omega)$.
