(a) Show that there must exist x,y∈[0,1] satisfying |x−y|=12 and f(x)=f(y)
I can start by defining a function g(x)=f(x+12)−f(x) to guarantee an x,y so that |x−y|=12 But how do I show that f(x)=f(y)?
(b) Show that for each n∈N ∃xn,yn∈[0,1] with |xn−yn|=1n, and f(xn)=f(yn)
Actually I'm not sure where to start here. Any help is greatly appreciated.
Thanks!
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