Question

Still can't get this one:
Suppose that f is continuous on [0,1] and that f(0) = f(1). Show that there is a point \[c \in [0,1/2]\]such that f(c) = f(c+1/2).

Answer 1

I'm guessing intermediate value theorem is needed here.

Answer 2

here is a thought
consider the function \(g(x)=f(x)-f(x+\frac{1}{2})\) which must be continuous because \(f\) is
then \(g(0)=f(0)-f(\frac{1}{2})\) and
\(g(\frac{1}{2})=f(\frac{1}{2})-f(1)=f(\frac{1}{2})-f(0)=-g(0)\)

Answer 3

if they are both 0 then you are done, and if one is positive the other must be negative, so by IVT you know it has to be equal to 0 somewhere