Finite and Infinite support in time and frequency domain
The sinc is just fine an example as a signal with infinite support: 

Support being defined as the smallest interval in which the function has non-zero values, it's trivial to see that $\text{sinc}(x)=\sin(x)/x$ (for all $x$ but a single point) becomes arbitrarily small for large $x$, but never actually constantly $\text{sinc}(x&gt;\xi)=0$ for any finite $\xi$. So, q.e.d., sinc has infinite support.

We use this exact property to illustrate to students all over the world that the perfect filter (i.e. something that looks like a rectangle in frequency domain) can never be implemented as a FIR filter, as that would need to be infinitely long.