Let $$\mathcal{H}^{1,1} (T^1 )$$ denote the Hardy space of real-valued functions on the unit circle with weak derivatives in the usual real Hardy space $$\mathcal{H}^1 (T^1 )$$ . It is shown that when the weak derivative of a locally Lipschitz continuous function$f$has bounded variation on compact sets the Nemytskii operator$F$, defined by$F(u)=f·u$, maps $$\mathcal{H}^{1,1} (T^1 )$$ continuously into itself. A further condition sufficient for the continuous Fréchet differentiability of$F$is then added.