I am given a function $f$ that is continuous and periodic with period $p$. The domain of the function is entire real numbers. I have to prove that the function has a global maximum.
Attempt: First, I proved that a continuous function that is bounded by two elements on its domain must have a maximum.
Question:
Is it correct to now just write that since $f$ is periodic, if I consider the function in an interval $[0,p]$, then the function will just be repeated outside the interval and since I have proved that there is a maximum in that interval, it will also be the global maximum? What I am confused about is: the question asks me to prove existence of a global maximum. But if the function is repeating with period $p$, there are many such maxima. But isn't global maximum unique? What am I understanding wrong?
From the comments above by @Furrane.
The global maximum is unique but it can be attained by multiple points. Take $f(x)=\sin(x)$. It is $2\pi$-periodic and has a global maximum equal to $1$. But, this maximum is attained by an infinite number of points, namely every point of the form $\frac{\pi}{2} + 2 k \pi$, where $k \in \mathbb{Z}$.