Suppose f : R → R is continuous and periodic with period 2a for some...

Nicholas Hunter

Nicholas Hunter

Answered

2022-11-23

Suppose f : R R is continuous and periodic with period 2a for some a > 0; that is, f ( x ) = f ( x + 2 a ) for all x R. Show there is some c [ 0 , a ] such that f ( c ) = f ( c + a ).
The only way I see to do this question is to apply the intermediate value theorem, but I just don't know how to apply it to this question. I know that f ( x ) = f ( x + 2 a ). Therefore, if i can show that f ( c + a ) = f ( c + 2 a ) or f ( c + a ) f ( c + 2 a ) = 0, I'd be done. But I just don't know where to go from there. I tried substituting x = a, getting f ( 2 a ) f ( 3 a ), but I can't show that's less than, equal to, or greater than 0. Any ideas?

Answer & Explanation

Kalmukujobvg

Kalmukujobvg

Expert

2022-11-24Added 14 answers

Hint: Define a function g ( x ) = f ( x + a ) f ( x ). If g ( 0 ) = 0, then you're done; if not, note that g ( a ) = g ( 0 ).

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