# True/false : There exist $f : S^1 \rightarrow \mathbb{R}$ which is continuous and onto

Is the following statement is true/false ?

There exist $$f : S^1 \rightarrow \mathbb{R}$$ which is continuous and onto.

My thinking: yes, because for every function $$f : S^1 \rightarrow \mathbb{R}$$ there exist uncountably many pairs of distinct points $$x$$ and $$y$$ $$\in S^1$$ such that $$f(x) = f(y)$$.

• That's not what "onto" means. – Matthew Daly Aug 14 at 7:13

• In other words: $f$ has a minimum $m$ and a maximum $M$ on $S^1$, so that $f(S^1) \subset [m, M]$. – Martin R Aug 14 at 8:00