Suppose f is continuous on [0,2] and that f ( 0 ) = f (...

seupeljewj

seupeljewj

Answered

2022-06-24

Suppose f is continuous on [0,2] and that f ( 0 ) = f ( 2 ). Then x , y [ 0 , 2 ] such that | y x | = 1 and that f ( x ) = f ( y ).


Let g ( x ) = f ( x + 1 ) f ( x ) on [0,1]. Then g is continuous on [0,1], and hence g enjoys the intermediate value property! Now notice
g ( 0 ) = f ( 1 ) f ( 0 )
g ( 1 ) = f ( 2 ) f ( 1 )
Therefore
g ( 0 ) g ( 1 ) = ( f ( 0 ) f ( 1 ) ) 2 < 0
since f ( 0 ) = f ( 2 ). Therefore, there exists a point x in [0,1] such that g ( x ) = 0 by the intermediate value theorem. Now, if we pick y = x + 1, i think the problem is solved.

I would like to ask you guys for feedback. Is this solution correct? Is there a better way to solve this problem?

Answer & Explanation

alisonhleel3

alisonhleel3

Expert

2022-06-25Added 23 answers

It is enough to write g ( 1 ) = g ( 0 )..

And, as Hagen commented, you only forgot to mention the case when already f ( 0 ) = f ( 1 ), but then it is readily done.

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