Maximum between two consecutive minima, topology

Question

I couldn't fall nasleep tonight due to a math problem I made up.

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous fubction. Then, there is always a local maximum between two consecutive local minima.

My friends and I showed this this afternoon with the sturm liousville theorem, but I am quite sure there must be a topological argument as well, but I can not find it. I tried connectedness (since $f$) is continuous but It failed. Does anyone know a ("simple"), elegant topological approach to this problem?

Answer 1

Let $a< b$ be two points of local minima. Since the interval $[a,b]$ is a compact and connected set, its image under $f$ is compact and connected, thus an interval $[c, d]$. Moreover, since $a$ and $b$ are local minima, we have $f(a), f(b)< d$. Thus there is $p\in (a,b)$ such that $ f(p)= d$, which is clearly a local maximum.

Answer 2

This problem might need an adjustment to be well posed.

But the following should work:

If $c<c'$ then since the function $f$ is continuous and $[c,c']$ is compact $f$ achieve a maximum in $[c,c']$, let's call it $c''$. Suppose that $c$ and $c'$ are local minima. If $c''$ lies in the interior you are done, but if $c''=c'$ then that function is locally constant near $c'$ and again you are done.