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

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

Any help would be greatly appreciated. Thanks a lot!
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Good question! Try the function $g\left(x\right)=f\left(x+a\right)-f\left(x\right)$ 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