Plot of a DTFT is not periodic

1. Is the DTFT always periodic?
2. Why should continuous Fourier transform be Aperiodic whereas DTFT is periodic?
3. Does FFT assume periodicity?
4. Why is DTFT continuous?

Is the DTFT always periodic?

Sampling the DTFT always induces a periodic sequence in the time domain (in the same way that sampling x[n] always results in a periodic X(ejω)).

Why should continuous Fourier transform be Aperiodic whereas DTFT is periodic?

The spectrum of any discrete signal has a period of 2*pi. Thus, the fourier coefficients occur periodically at interval of 2*pi. While , in case of continuous time signal the spectrum has no such definite period.

Does FFT assume periodicity?

Originally Answered: How would one apply the FFT on a non-periodic signal? Discrete Fourier transform implicitly assumes that the signal is periodic, so you have to feed it something that you agree to consider periodic.

Why is DTFT continuous?

DTFT is continuous because the original time-domain signal that you sampled has well behaved transform which means it doesn't envolve impulses in the transformed domain (fourier domain). Because DTFT is just repeated version of the actual FT at every intervals. The DTFT itself is a continuous function of frequency.