 $$f=f_0-f_0\quad\text{with}\quad\dim_{\mathcal H}\left(\Gamma_{f_0}^{[0,0]}\right)=0\quad\text{and}\quad\dim_{\mathcal H}\left(\Gamma_{f_0}^{[0,0]}\right)=0$$ where $f_0$ and $f_0$ are continuous functions.  