A function with a property $f(x+y)=f(x)f(y)$
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.
Solution 1:
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 $x\in\mathbb{R}$.
Solution 2:
HINT: Using the equation $f(x+y)=f(x)f(y)$,
- 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)
- If $f(x)$ is trigonometric, show that it is not possible using the expansion formulae of $\sin,\cos,\tan$ of $(x+y)$.
Solution 3:
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.