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I'm working on the following question:

Which of the following three properties imply $f$ is monotone (it maybe none or more than 1):

1. $f(x + y) = f(x)f(y)$
2. $f(x - y) = f(x)f(y)$
3. $f$ is periodic and non-decreasing.

I think 3 works, I'm not sure how to evaluate 1 and 2.

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2 Answers

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(1) does not imply monotonicity. Take $g$ to be a discontinuous (also non-integrable, non-monotone) solution to Cauchy's Functional Equation:$$g(x + y) = g(x) + g(y).$$Then $f = \exp \circ g$ is a non-monotone function satisfying$$f(x + y) = f(x)f(y).$$

(2) For such a function $f$, first note that $f(0 - 0) = f(0)f(0)$, which implies $f(0) = 0$ or $f(0) = 1$. If $f(0) = 0$, then$$0 = f(x - x) = f(x)f(x) \implies f(x) = 0$$for all $x$, which tells us that $f$ is constant (and hence monotone). Otherwise, $f(0) = 1$, and similarly we see that $f(x) = \pm 1$ for all $x$. The only monotone functions possible are constant. So, the question is, are there any non-constant solutions?

Suppose $f$ is a non-zero solution. I claim that $f$ is a homomorphism from the group $(\mathbb{R}, +)$ into the group $(\{-1, 1\}, \cdot)$. If $x, y \in \mathbb{R}$, then$$f(y) = f(x + y - x) = f(x + y)f(x) \implies f(x + y) = f(y) f(x)^{-1} = f(x) f(y).$$If $f$ is not constantly $1$, then $f$ maps onto $\lbrace -1, 1\rbrace$, and $f^{-1}\lbrace 1 \rbrace$ is a group of index $2$ in $(\mathbb{R}, +)$, which doesn't exist. Thus, the only solutions are constant solutions $0$ and $1$, both of which are monotone. So yes, (2) implies monotonicity.

(3) This also implies monotonicity, as it specifies non-decreasing. Such functions are also constant, making me wonder if the question was about the functions being constant, not monotone?

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The first one should remind you of the exponential function $e^{x+y}=e^xe^y$, together with that, what are some things you must know about the function? What is $f(x+0)$? $f(x/2+x/2)$? $f(nx)$?

Is the second one even, odd, neither? What can be said about the shapes of such functions?

What family of functions does the third one belong to? How can something never decrease but still be periodic? This is a very strict requirement.

That should provide enough guidance to get you off the ground. Remember when posting questions in the future you are supposed to show that you've spent time researching it, and show your effort so far.

Good luck!

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