Let $f : S^2 \to \mathbb{R}$ be a continuous map such that $f(-x) = -f(x)$. Consider the set $Z = f^{-1}(0)$. Must $Z$ contain some path from some point to its antipode? Indeed, must $Z$ contain a continuous loop intersecting each "meridian", passing through antipodal points on antipodal meridians?

[I see this often asserted as a step in intuitive arguments for the Borsuk-Ulam theorem, but do not see why rigorously it must be so. Indeed, as a child, I once attempted precisely this argument for the Borsuk-Ulam theorem in a math camp, and was chided for asserting this unsupported lemma; thus it has stuck with me always. Certainly, $Z$ must contain a point on each meridian, but that these points must line up into a continuous path is not obvious to me.]

Update: My original question turned out to have already been answered at [How bogus is the glitzy proof of Borsuk-Ulam?](https://mathoverflow.net/questions/251921/how-bogus-is-the-glitzy-proof-of-borsuk-ulam), as well as by Loïc Teyssier in the same way below, but a new question with same motivation is: must Z contain two antipodal points in the same connected component? (If so, then the Borsuk-Ulam proof via this approach can still be salvaged; if not, we may write it off as a lost cause)