Lemma 12.4 (The lifting lemma):

Let $X$ be a compact subset of $\mathbb{R}^{n}$ that is star shaped with respect to origin. Let $f : X \longrightarrow S^1$ be a continuous function such that $f(0) =$   ex$(t_0)$ for some $t_0 \in \mathbb{R}.$ Then, there exists a continuous function $\tilde{f} : X \longrightarrow \mathbb{R}$ such that

   ex$$\tilde{f}(x) = f(x),\quad {\tilde f}(0) = t_0 \eqno(12.3)$$

Moreover the function $\tilde{f}$ satisfying (12.3) is unique and is called the lift of $f$ with respect to ex.