  In this paper we study the $c$-function of the sine-Gordon model taking
explicitly into account the periodicity of the interaction potential. The
integration of the $c$-function along trajectories of the non-perturbative
renormalization group flow gives access to the central charges of the model in
the fixed points. The results at vanishing frequency $\beta^2$, where the
periodicity does not play a role, are retrieved and the independence on the
cutoff regulator for small frequencies is discussed. Our findings show that the
central charge obtained integrating the trajectories starting from the
repulsive low-frequencies fixed points ($\beta^2 <8\pi$) to the infrared limit
is in good quantitative agreement with the expected $\Delta c=1$ result. The
behavior of the $c$-function in the other parts of the flow diagram is also
discussed. Finally, we point out that also including higher harmonics in the
renormalization group treatment at the level of local potential approximation
is not sufficient to give reasonable results, even if the periodicity is taken
into account. Rather, incorporating the wave-function renormalization (i. e.
going beyond local potential approximation) is crucial to get sensible results
even when a single frequency is used.
