# Linked Questions

3answers
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### 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 ...
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 ...
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 ...
3answers
4k views

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 $g(xy) = g(x) + ... 1answer 2k views ### Overview of basic facts about Cauchy functional equation The Cauchy functional equation asks about functions$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 ... 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 ... 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 ... 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. 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$f(x)$... 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$c \in \mathbb{R}$... 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 ... 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$is ... 1answer 717 views ### continuous functions on$\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$... 1answer 416 views ### Function of sum -> Product of functions, sum of functions I am looking for a set of functions$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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