I have just recently covered the Intermediate Value Theorem, and I wanted to practice solving proble

Dashawn Robbins 2022-05-03 Answered
I have just recently covered the Intermediate Value Theorem, and I wanted to practice solving problems involving this theorem. However, I encountered a problem that I am not exactly sure how to tackle (it's a question involving a periodic function).

I thought about splitting the proofs into 3 cases, but I don't think it would be applicable here?

The question is:

If f is periodic with a period of 2a for some a > 0, then f ( x ) = f ( x + 2 a ) for all x R. Show that if f is continous, there exists some c [ 0 , a ] such that f ( c ) = f ( c + a ).

Any help would be greatly appreciated. Thanks a lot!
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Answers (1)

hadnya1qd
Answered 2022-05-04 Author has 15 answers
Good question! Try the function g ( x ) = f ( x + a ) f ( x ) since the sum/difference of continuous functions is continuous, g is continuous. If you still can't get it, leave a comment for me. note this may not work, but on first look I am pretty sure it will
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So I'm thinking about applying the intermediate value theorem:

If
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But I couldn't think of any way to prove that f ( a ) < f ( x k ) < f ( b ) or is it even true?

EDIT: Thanks everyone for your effort.