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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?

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    Your phrasing is unclear. Are you requesting a characterization of the functions $f$ for which there exist $a<b<0<c<d$ such that $f(a) f(b)$ and $f(c) f(d)$ are negative? Or maybe something like $g(x)=\sin(1/x)$, which intersects every (axis-aligned) rectangle containing the origin? – Paul Tanenbaum Nov 01 '23 at 12:04
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    You might look at https://math.stackexchange.com/questions/3074999/function-whose-graph-is-dense-in-the-plane for functions whose graph intersects every rectangle. – Gerry Myerson Nov 01 '23 at 12:10
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    The exponential function does not satisfy $f(x+y)=f(x)+f(y)$; it satisfies $f(x+y)=f(x)f(y)$. – Gonçalo Nov 01 '23 at 12:23
  • @Gonçalo Yes i edited. Thank you –  Nov 01 '23 at 12:47
  • @PaulTanenbaum not the text says that a function defined in that way there are infinite ones. Furthermore, there are infinite rectangles that contain the graph. But in the piano in general not only in the origin. –  Nov 01 '23 at 13:58

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