If fff is a continuous function on [0,1][0,1][0,1] such that f(0)=f(1)f(0)=f(1)f(0)=f(1), must there exist c∈[0,1/2]c \in [0, 1/2]c∈[0,1/2] such that f(c)=f(c+1/2)f(c) = f(c + 1/2)f(c)=f(c+1/2)?
Yes, by the Intermediate Value Theorem
No, this is not guaranteed
Only if fff is differentiable
Only if fff is monotonic