Let $f\colon \mathbb{S}^{1}\to\mathbb{S}^{1}$ be a continuous function of degree 1 and $F\colon \mathbb{R}\to \mathbb{R}$ a lift of $f.$ When $f$ is a orientation-preserving homeomorphism, the limit $$\tau(y)=\lim_{n\to \infty}\frac{F^{n}(y)-y}{n} $$ exists for every $y,$ it doesn't depend on $y$ and is called the rotation number of $F.$ Also, if we ask $f$ to be only continuous but do require $F$ to be non-decreasing, we still have the result.
On the other hand, if we do not make such requirements for $F$ we may have $F(y)=y+\sin(2\pi y)$ and then we no longer have independence on $y.$
My question is: what about existence? Are there any examples such that $F$ is a lift for a continuous circle map of degree one $f$ but there is some $y_{0}\in \mathbb{R}$ such that $\tau(y_{0})$ is not defined?