Dashawn Robbins

2022-05-03

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 $f(x)=f(x+2a)$ for all $x\in R$. Show that if f is continous, there exists some $c\in [0,a]$ such that $f(c)=f(c+a)$.

Any help would be greatly appreciated. Thanks a lot!

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 $f(x)=f(x+2a)$ for all $x\in R$. Show that if f is continous, there exists some $c\in [0,a]$ such that $f(c)=f(c+a)$.

Any help would be greatly appreciated. Thanks a lot!

hadnya1qd

Beginner2022-05-04Added 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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