Given a continuous function $f: \mathcal{R} \rightarrow \mathcal{R}$ with:
$$\lim_{x \rightarrow \infty}f(x) = 0$$
$$\lim_{x \rightarrow -\infty}f(x) = 0$$
$$f(x)>0 \, \forall \, x \in \mathcal{R}$$
Show that there exists $x,y \in \mathcal{R}$ with $x\neq y$ so that $f(x)=f(y)$.
In this question no interval is given. What I want to do is to show that there is a maximum and then use the Intermediate value theorem. But I don't know how to do that without any interval given.
Is it sufficient enough to say that because the limit for x to infinity and -infinity and f(x)>0 there must exist a maximum value? Since f(x) cannot be zero.
Hope to see your suggestions.
Best Answer
HINT: Take any $x_0$ to start. Then $f(x_0)>0$ and $\lim_{x \rightarrow -\infty} f(x) = \lim_{x \rightarrow \infty} f(x) = 0$ imply that
there is an $x<x_0$ such that $f(x) = \frac{f(x_0)}{2}$, because as $x$ goes from $x_0$ to $-\infty$ and $f(x)$ vanishes, the value of $f(x)$ must pass through $\epsilon$, for each $\epsilon$ satisfying $f(x_0)>\epsilon>0$.
there is a $y>x_0$ such that $f(y) = \frac{f(x_0)}{2}$.
There is your $x,y$.