Let $f : (0, ∞) → \mathbb{R}$ be a function for which $[f(x) · f(1/x)]<0$, for every $x>0$ where $x$ is not equal to $1$.
Suppose that $f(x)$ is continuous at $x'=1$, then $f(1) = 0$.
Notice that according to the given information, $f(x)>0$ and $f(1/x)<0$ (or the other way around).
I'm trying to figure out if the above statement is true or false, but I have mixed feelings and I'm not sure if it's false or if it's true and there is something that I'm missing.
According to the intermediate-value theorem, if it was given that $f$ is continuous for every $x$ in $\mathbb{R}$ or even (just as an example) every $x$ in $(0,2)$, then no problem. But it's not given and this is all the information given to me, and I can't be sure if this means that it's false or not.
I also can't think of a solid way of disproving the statement and therefore I'm stuck.
I would appreciate any suggestions!