Linked Questions

6
votes
3answers
799 views

Classifying Functions of the form $f(x+y)=f(x)f(y)$ [duplicate]

Possible Duplicate: Is there a name for such kind of function? The question is: is there a nice characterization of all nonnegative functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that ...
2
votes
1answer
462 views

What can we say about functions satisfying $f(a + b) = f(a)f(b) $ for all $a,b\in \mathbb{R}$? [duplicate]

Possible Duplicate: Is there a name for such kind of function? I am investigating functions satisfying the exponentiation identity $f(a + b) = f(a)f(b)$ for all $a,b\in \Bbb R$. This is ...
0
votes
2answers
139 views

Is $f(x)f(y)=f(x+y)$ enough to determin $f$? [duplicate]

I had a discussion with a friend and there it came up the question whether $f(x)f(y)=f(x+y)$, $f(0)=1$ and the existence of $f'(x)$ implies that $f(x)=\exp(a x)$. This seems very reasonable but I ...
9
votes
3answers
4k views

If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t

Let $f(xy) =f(x)f(y)$ for all $x,y\geq 0$. Show that $f(x) = x^p$ for some $p$. I am not very experienced with proof. If we let $g(x)=\log (f(x))$ then this is the same as $g(xy) = g(x) + ...
17
votes
1answer
2k views

Overview of basic facts about Cauchy functional equation

The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of ...
5
votes
2answers
539 views

Functional Equation $f(x+y)=f(x)+f(y)+f(x)f(y)$

I need to find all the continuous functions from $\mathbb R\rightarrow \mathbb R$ such that $f(x+y)=f(x)+f(y)+f(x)f(y)$. I know, what I assume to be, the general way to attempt these problems, but I ...
14
votes
4answers
475 views

Constant functions

Let $f$ , $g$ , $h$ be three functions from the set of positive real numbers to itself satisfying $$f(x)g(y) = h\left((x^2+y^2)^{\frac{1}{2}}\right)$$ for all positive real numbers $x$ , $y$ . Show ...
4
votes
3answers
1k views

Functional equation $f(xy)=f(x)+f(y)$

I want to prove the following claim: If $f:(0,\infty)\to\mathbb{R}$ satisfying $f(xy)=f(x)+f(y)$, and if $f$ differentiable on $x_0=1$, then $f$ differentiable for all $x_0>0$. Thank you.
6
votes
5answers
154 views

How prove $f(x)$ is a monotonic function if $f(x+y)=f(x)f(y)$

Let $f(x)$ be a real valued, differentiable function such that for any $x,y \in \mathbb{R}$,$f(x+y)=f(x)f(y)$. Suppose there exist $a,b$ such that $f(a)\neq 0, f'(b)>0$. Show that $f(x)$ ...
14
votes
2answers
807 views

If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere

Prove that if $f\colon\mathbb{R}\to\mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ for all $x,y$, and $f$ is continuous at $0$, then it is continuous everywhere. If there exists $c \in \mathbb{R}$ ...
4
votes
2answers
581 views

Differentiable function, not constant, $f(x+y)=f(x)f(y)$, $f'(0)=2$

Let $f: \mathbb R\rightarrow \mathbb R$ a derivable function but not zero, such that $f'(0) = 2$ and $$ f(x+y)= f(x)\cdot \ f(y)$$ for all $x$ and $y$ belongs $\mathbb R$. Find $f$. My first ...
3
votes
3answers
361 views

Locally Bounded Functional Equation $f(x+y) = f(x) + f(y)$ and Continuity

Let $f$ be a real-valued function on $\mathbb{R}$ s.t. $f(x+y) = f(x) + f(y)$ for all $x,y$ reals. Suppose there are reals $c$ and $M$ s.t. $|f(x)| \leq M $ for all $x$ in $[-c,c]$. Show that $f$ is ...
1
vote
1answer
717 views

continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$

Let $g$ be a function on $\mathbb R$ to $\mathbb R$ which is not identically zero and which satisfies the equation $g(x+y)=g(x)g(y)$ for $x$,$y$ in $\mathbb R$. $g(0)=1$. If $a=g(1)$,then $a>0$ ...
0
votes
1answer
416 views

Function of sum -> Product of functions, sum of functions

I am looking for a set of functions $f\left(\sum_i x_i\right)$ that satisfies either or both of the following properties ($x_i\in\mathbb{R}^+$): $f\left(\sum_i x_i\right)=\prod_i g\left(x_i\right)$ ...

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