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A function with the property $f(x+y)=f(x)f(y)$ is well known exponential function, $f(x)=a^x$. My question is, how do you prove if there is no other function with this kind of property?

Edit: I always find this in math contests. At first glance, it really is the exponential function. As I see in the comments section, there are many other functions.

cgo
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5 Answers5

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The comments that suggest $f(x)=1$ and $f(x)=0$ as solutions are correct, but these are just special cases of $f(x)=a^x$ with $a=1$ or $a=0$. If $f$ is not continuous then $f$ need not be an exponential function on the irrationals, but on $\mathbb{Q}$ $f(x)=a^x$ if it satisfies this property.

To see this, let $f(1)=a$ and note that $f(n+1)=f(n)f(1)$ so that by induction $f(x)=a^n$ for all $a\in\mathbb{N}$. Assuming that $a\neq0$, from $f(x+(-x))=f(x)f(-x)$ where $f(x+(-x))=f(0)=1$ (the fact that $f(0)=1$ must be proven first from consideration of $f(1+0)$), we get that $f(x)=a^x$ for all $x\in\mathbb{Z}$. We also have that $f(\underbrace{\frac{1}{p}+\ldots+\frac{1}{p}}_\text{$p$ terms})=(f(\frac{1}{p}))^p=f(1)=a$ so that $f(\frac{1}{p})=a^\frac{1}{p}$ so it's easy to argue from this fact that $f(x)=a^x$ for all $x\in\mathbb{Q}$.
Now if we know that $f$ is continuous everywhere (I believe it is also sufficient for $f$ to be continuous merely at one point) it can be shown that $f(x)=a^x$ for all $x\in\mathbb{R}$.

user293794
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    Assuming $f(x)$ is continuous, we could use this result: http://math.stackexchange.com/questions/505/can-there-be-two-distinct-continuous-functions-that-are-equal-at-all-rationals – cr001 Nov 27 '15 at 06:17
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The condition $f(x+y)=f(x)f(y)$ only implies $f(x)=a^x$ for all rational numbers $x\in\mathbb{Q}$ and for some $a\in\mathbb{R}$. You can get this equality for all real numbers if you have more conditions, for example, if $f$ is continuous in $\mathbb{R}$ or if $f$ is Lebesgue-measurable.

Ángel Valencia
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I proved at Proof of existence of $e^x$ and its properties that, if $f(x)$ is differentiable at $0$, then $f(x+y) =f(x)f(y) $ implies that $f'(x) =f'(0) f(x) $.

This leads immediately to the exponential or power function.

marty cohen
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  • It holds if $f(0)=1$. – Empty Nov 27 '15 at 06:49
  • It would, assuming that $f$ is continuous and with this condition $(f(x+y)=f(x)f(y)) $ and $f(0)=1$ to prove that $\displaystyle\lim_{x\to-\infty} f(x)=0$ ?

    My sketch As $f$ is continuous, it suffices to consider a sequence $(a_n)n$ with limit $-\infty$ and prove that $\displaystyle\lim{n\to+\infty} f(a_n)=0$. This is true ? Because then the only option I see is to prove that this is the exponential function and then conclude that the limit is $0$.

     – Pierre Aug 15 '22 at 15:11
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It would, assuming that $f$ is continuous and with this condition $(f(x+y)=f(x)f(y)) $ and $f(0)=1$ to prove that $\displaystyle\lim_{x\to-\infty} f(x)=0$ ?

My sketch

As $f$ is continuous, it suffices to consider a sequence $(a_n)_n$ with limit $-\infty$ and prove that $\displaystyle\lim_{n\to+\infty} f(a_n)=0$. This is true ? Because then the only option I see is to prove that this is the exponential function and then conclude that the limit is $0$.

Pierre
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HINT: Using the equation $f(x+y)=f(x)f(y)$,

  1. If $f(x)$ is polynomial, show that the degree of L.H.S. $\not =$ degree of R.H.S. (except if $f(x)$ is the zero polynomial)
  2. If $f(x)$ is trigonometric, show that it is not possible using the expansion formulae of $\sin,\cos,\tan$ of $(x+y)$.
SchrodingersCat
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