Let f be a continuous function on R which is periodic with period 2 π. This means f ( t + 2 π ) = f ( t ) for all t. Show that there exists x ∈ [ 0 , π ] such that f ( x ) = f ( x + π ).I know it's an intermediate-value theorem problem. I think I have to take the difference of both sides, but not quite sure. Can anyone help?

Question

Let   f be a continuous function on       R   which is periodic with period   2  π. This means   f  (  t  +  2  π  )  =  f  (  t  ) for all   t. Show that there exists   x  ∈  [  0  ,  π  ] such that   f  (  x  )  =  f  (  x  +  π  ).I know it&#039;s an intermediate-value theorem problem. I think I have to take the difference of both sides, but not quite sure. Can anyone help?

radcas87gex5r · Accepted Answer

Consider   g  (  x  )  =  f  (  x  +  π  )  −  f  (  x  ),   g  (  0  )  =  −  g  (  π  ). If   g  (  0  )  =  0 we&#039;re done (why ?), otherwise   g has a root (why ?) and we&#039;re also done (why ?).

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