Uniqueness of exceptional function

Question

I was reading some note of uniqueness of the exponential function.
For every $a>0$ exists a unique $f: \mathbb{R} \to (0,\infty)$ such that
(1) $f(x+y)=f(x)f(y)$
(2) $f(1)=a$
(3) exists a rectangle in a the first or second quadrant that not contains the graphs.
The question is:
Which are the function that have a points with every rectangle in the first or second quadrant of the plane xy?