Saturday, May 26, 2018

real analysis - Let f:[0,1]rightarrowmathbbR be continuous with f(0)=f(1) *note, there is a part b*

(a) Show that there must exist x,y[0,1] satisfying |xy|=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 |xy|=12 But how do I show that f(x)=f(y)?



(b) Show that for each nN xn,yn[0,1] with |xnyn|=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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