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

aurelegena 2022-09-06 Answered
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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Answers (1)

farbhas3t
Answered 2022-09-07 Author has 6 answers
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
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