Show that if f is continuous on [0,1] with f ( 0 ) = f ( 1 ), there must exist x , y ∈ [ 0 , 1 ] with | x − y | = 1 2 and f ( x ) = f ( y )I've been working on this for a while, and can't seem to figure out where to start. Any hints would be appreciated

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Show that if   f is continuous on [0,1] with   f  (  0  )  =  f  (  1  ), there must exist   x  ,  y  ∈  [  0  ,  1  ] with       |    x  −  y      |    =      1    2   and   f  (  x  )  =  f  (  y  )I&#039;ve been working on this for a while, and can&#039;t seem to figure out where to start. Any hints would be appreciated

farbhas3t · Accepted Answer

Let   g be the function defined at   [  0  ,      1    2    ] by  g  :  t  ↦  f  (  t  )  −  f  (  t  +      1    2    )we have  gg is continuous at   [  0  ,      1    2    ]and  g  (  0  )  .  g  (      1    2    )  =  −  (  f  (  0  )  −  f  (      1    2    )      )    2    ≤  0 since   f  (  0  )  =  f  (  1  ).then  ∃  x  ∈  [  0  ,      1    2    ]   such that   g  (  x  )  =  0 or  f  (  x  )  =  f  (  x  +      1    2    )  =  f  (  y  )with   y  =  x  +      1    2   satisfying      |    y  −  x      |    =      1    2

Show that if f is continuous on [0,1] with f(0)=f(1), there must exist x,y in [0,1] with |x−y|=1/2 and f(x)=f(y) I've been working on this for a while, and can't seem to figure out where to start. Any hints would be appreciated

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