BIBO Stability and the convergence of the frequency response of a system
The same happens with sinusoids: they have a Fourier transform even though they are not absolutely summable in the time domain. These are cases in which the Fourier transform doesn't converge uniformly, but it does converge in some other sense.

The signal $x(n)=\mathrm{sinc}(n)$ is not absolutely summable... but it is square-summable (i.e. it has finite energy). Therefore, the Fourier transform will converge but in the mean-square sense. In these cases, the transform converges to a discontinuous function (in the case of this specific signal, to a rectangular pulse).

Similarly, sequences that involve sinusoids like $x(n)=\sin(\omega_0n)$ are neither absolutely nor square summable, but a Fourier transform can be defined using distributions (Dirac delta functions).

In both cases the Fourier transforms are not continuous. They are not infinitely differentiable either.

As a general rule we can state that if some impulse response $h(n)\in l^1(\mathbb{Z})$, then its Fourier transform will be continuous. If the Fourier transform is not continuous, then you know that its inverse is not in the sequence space $l^1$.

You mentioned BIBO stability, so I think this may be of interest. There is a theorem that states:


  An LTI system with impulse response $h(n)$ is BIBO stable if and only if $h(n)\in l^1(\mathbb{Z})$.


So one can judge the stability of a system by looking at the continuity of its transfer function.