Lift of a function f of the torus

Question

I have defined a lift of a function $f: \mathbb{R}/\mathbb{Z} \rightarrow \mathbb{R}/\mathbb{Z}$ as $\hat{f}(t)=f_0+\int_0^t f'([s]) ds$, where $f_0$ is the lift to $\mathbb{R}$ of $f([0])$ and $f'\in\mathbb{R}$ is well defined taking the same lift in the definition $$ f'(\theta)=\lim_{\epsilon \rightarrow 0} \frac{\hat{f}(\theta+\epsilon)-\hat{f}(\theta)}{\epsilon}. $$ Any idea of how to generalize this definition of the lift and derivative to the case $f:\mathbb{R}^n/\mathbb{Z}^n \rightarrow \mathbb{R}^n/\mathbb{Z}^n$? Thank you!