Let S denote a unit circle centered at origin in xy plane, and
f is a continuous function that sends S to R (no need to be 1 to 1 or onto).
show that there's (x, y) such that f(x, y) = f(-x, -y)
have a feeling it has something to do with theorems related to topology.
The Attempt at a Solution
by re-writing S as (cos z, sin z) for 0 <= z < 2 pi
i can reduce f to a function of one variable, f(z), such that
f(z) = - f(z + pi), but then I'm stuck what to do after that.
(or maybe I shouldn't have reduced f to a function of one variable in the first place?)