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I was wondering if there is a name for a function that satisfies the conditions

$f:\mathbb{R} \to \mathbb{R}$ and $f(x+y)=f(x) \cdot f(y)$?

Thanks and regards!

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3 Answers 3

up vote 46 down vote accepted

If $f(x_0)= 0$ for some $x_0\in\mathbb{R}$, then for all $x\in\mathbb{R}$, $f(x)=f(x_0+(x-x_0))=f(x_0)\cdot f(x-x_0)=0$. Therefore, either $f$ is identically $0$ or never $0$. If $f$ is not $0$, then it is a homomorphism from the group $\mathbb{R}$ with addition to the group $\mathbb{R}\setminus\{0\}$ with multiplication. If $f(x)<0$ for some $x$, then $f(\frac{x}{2})^2\lt 0$, which is impossible, so $f$ is actually a homomorphism into the positive real numbers with multiplication. By composing with the isomorphism $\log:(0,\infty)\to\mathbb{R}$, such $f$ can be analyzed by first analyzing all additive maps on $\mathbb{R}$. Assuming continuity, these all have the form $x\mapsto cx$ for some $c\in \mathbb{R}$, and hence $f(x)=\exp(cx)$. Assuming the axiom of choice, there are discontinuous additive functions on $\mathbb{R}$ that can be constructed using a Hamel basis for $\mathbb{R}$ over $\mathbb{Q}$, and thus there are also discontinuous homomorphisms from $\mathbb{R}$ to $(0,\infty)$.

So for an actual answer to the question: Yes, they are called (the zero map or) homomorphisms from the additive group of real numbers to the multiplicative group of positive real numbers.

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+1 for complete solution :-) –  Aryabhata Feb 15 '11 at 0:24

If you enforce continuity on $f(x)$ (continuity even at a single point should do), then the only function which satisfies this is an exponential function of the form $f(x) = e^{ax}$.

You need to enforce continuity to extend this from rationals to reals (as you would expect).

If you have $f: \mathbb{Q} \rightarrow \mathbb{R}$, then $f(x) = e^{ax}$ is the only function.

(The proof for this is analogous to proving $f(x) = cx$ when $f(x+y) = f(x) + f(y)$)

In fact one of the definitions of $f(x) = a^x$ is as follows. If there exists a function $f(x)$ satisfying the following properties,

  1. $f(x) : \mathbb{R} \rightarrow \mathbb{R}$

  2. $f(x+y) = f(x) \times f(y)$

  3. $f(x) \text{ is continuous at-least at one point in } \mathbb{R}$

  4. $f(1) = a$

Then, $f(x) = a^x$.

Note: $e^x$ could be defined like above with $f(1) = e$, where $\displaystyle e = \lim_{n \rightarrow \infty} \left( 1+\frac{1}{n} \right)^n$. This is one of the many definitions for $e^x$.

EDIT:

Without continuity you could define a function as follows:

We know that along all rationals $f(x) = e^{ax}$ where $x \in \mathbb{Q}$. Now lets define $f(x)$ over $x \in \mathbb{R} \backslash \{ \mathbb{Q} \}$.

For this, consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$. This is an infinite-dimensional vector space and the axiom of choice guarantees the existence of a basis for this space.

Now consider a basis for this space. Let the set of basis be $\{h_{\alpha}\}$ i.e. every real number can now be written as $r = \displaystyle \sum_{i=1}^{N(r)} q_i h_{\alpha_i}$, where $q_i \in \mathbb{Q}$, $N(r) \in \mathbb{N}$, and $h_{\alpha_i} \in \{h_{\alpha}\}$.

Now you have complete freedom to decide what the value of $f$ should take on each element in the set $\{ h_{\alpha} \}$.

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Thanks! What if $f$ can be either continuous or continuous? Are there other functions besides the exponential function that can satisfy the condition? –  Tim Feb 15 '11 at 0:48
    
@Tim: With the axiom of choice, there are other examples (as I mentioned in my answer). It is consistent with Zermelo Fraenkel set theory without choice that there are no discontinuous examples. –  Jonas Meyer Feb 15 '11 at 0:51
    
@Tim: If the function is not continuous anywhere, then you could have infinitely many functions. I will edit my post to explain it in detail. –  user17762 Feb 15 '11 at 0:52
    
@Sivaram: You seem to be claiming that $S_1\cup\{1\}$ is linearly independent over $\mathbb{Q}$. But this is not true, because for example the representatives of $\sqrt 2$, $\sqrt 3$, and $\sqrt 2+\sqrt 3$ are all distinct. You could invoke the existence of a Hamel basis in order to define discontinuous additive maps on $\mathbb{R}$, then take $\exp$ of these. –  Jonas Meyer Feb 15 '11 at 1:39
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@Sivaram: "You have complete freedom to decide what the value of $f$ should take on each element in $S_1$." What I'm saying is that for example if $s\in S_1$ is the representative of $\sqrt{2}+\sqrt{3}$, then $\sqrt{2}+\sqrt{3}=s+r$, $\sqrt{2}=s_2+r_2$, and $\sqrt{3}=s_3+r_3$ for some $r,r_2,r_3\in\mathbb{Q}$, and $s,s_2,s_3$ are distinct. $f(\sqrt{2}+\sqrt{3})=f(\sqrt{2})f(\sqrt{3})=f(s_2)f(r_2)f(s_3)f(r_3)$, so $f(s)=f(s_2)f(r_2)f(s_3)f(r_3)/f(r)$ is determined. –  Jonas Meyer Feb 15 '11 at 1:51

$\log f$ satisfies Cauchy's functional equation.

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You also need to show why $\log f$ is well-defined.(See Jonas' answer). +1 anyway. –  Aryabhata Feb 15 '11 at 0:18

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