A Version of The Implicit Function Theorem

Question

I want to ask if this version of the "implicit function theorem" holds.

Let $f:S^1\times S^1\to\mathbb{R}$ be continuous, for every $x\in S^1$, there exists $y\in S^1$, not necessarily unique, such that $f(x,y)=0$.

Does there exists $g:S^1\to S^1$, $g$ continuous, $f(x,g(x))=0$?

If this holds, is there an elementary proof for this?

Note that I did not include any constraints on differentiability.

Thanks.