Is this signal band-limited?

Question

I'm self-learning signal processing now, and I've run into this question about band-limited signals:

Consider the signal $x(t) = 1$ for $0 \leq t \leq T$ and $0$ otherwise. I've found that its Fourier transform is: $$ x_\phi(\Omega) = \frac{1-e^{-j\Omega T}}{j\Omega} $$

How can I determine if this signal is band-limited signal, or not?

Thanks in advance

Answer 1

Check if there is some $R>0$ such that $x_\phi (\Omega)=0$ holds for all $|\Omega |\geq R$. That would be the definition.

Another way is the following: Every unlimited signal is smooth (infinitely differentiable), even analytic by the Palye Wiener Theorem. Is this the case for your signal? (Is your signal continuous?)