Linked Questions

3
votes
2answers
468 views

Functions over $R$ such that $f(xy) = f(x)f(y)$ [duplicate]

Possible Duplicate: If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t Solution for exponential function's functional equation by using a definition of derivative I can think of ...
1
vote
6answers
440 views

Solutions of $f(x)\cdot f(y)=f(x\cdot y)$ [duplicate]

Can anyone give me a classification of the real functions of one variable such that $f(x)f(y)=f(xy)$? I have searched the web, but I haven't found any text that discusses my question. Answers and/or ...
2
votes
3answers
167 views

How can we find functions that satisfy $f(x\cdot y)=f(x)f(y)$? [duplicate]

Today I've encountered a question like The following; If function $f$ satisfies $f(xy)=f(x)f(y)$ and $f(81)=3$ then find The value of $f(2)$? What baffles me about this question is that I have to ...
1
vote
2answers
549 views

Find all multiplicative continuous functions on $(0,\infty)$ [duplicate]

If $x>0,y>0$ and if $f(xy)=f(x)f(y)$, then $f=\, ?$ I tried the problem. And got it as $f(x)^n=f(x^n)$. But answer is $f(x)=x^n$. How? $f$ is a continuous function.
11
votes
1answer
192 views

Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ satisfying a functional equation [duplicate]

Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ satisfying: $f\left ( x \right )f\left ( y \right )+ f\left ( xy \right )+ f\left ( x \right )+f\left ( y \right )= f\left ( x+y \right )+ 2\,...
2
votes
1answer
333 views

Multiplicative function on rationals [duplicate]

Let $\Bbb Q^+$ be the set of positive rational numbers. Find all solutions $f:\Bbb Q^+ \to \Bbb R$ of the functional equation $$ f(xy)=f(x)f(y), \quad x, y\in \Bbb Q. $$ Is $f(x)=x^a$ the only ...
1
vote
3answers
155 views

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be multiplicative. Is it exponential? [duplicate]

For function $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfies $f\left(x+y\right)=f\left(x\right)f\left(y\right)$ and is not the zero-function I can prove that $f\left(1\right)>0$ and $f\left(x\...
0
votes
2answers
150 views

The functional equation $f(xy)=f(x)f(y)$ [duplicate]

Let $f(x)$ be a function that satisfies this functional equation, $f(xy)=f(x)f(y)$. With a little bit of intuition and luck one may come to a conclusion that these are perhaps the solutions of $f(x)$,...
0
votes
0answers
19 views

prove $f(x)f(y)=f(xy), f(1)=1 \iff f(x)=x^k (k\ real)$ [duplicate]

prove $f(x)f(y)=f(xy), f(1)=1 \iff f(x)=x^k (k\ real)$ for $f:\mathbb{R}^+\to \mathbb{R}^+$ I find $f(a^r)=f(a)^r$ for rational r, but I cannot move to the next step.
86
votes
1answer
13k 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 ...
23
votes
5answers
6k views

Is there a name for function with the exponential property $f(x+y)=f(x) \cdot f(y)$?

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!
10
votes
2answers
5k 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 ...
6
votes
3answers
2k 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 $f(x+...
7
votes
3answers
3k views

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

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.
2
votes
2answers
1k views

If $f(x.y)=f(x).f(y)$ for all $x,y$ and $f(x)$ is continuous at $x=1$,then show that $f(x)$ is continuous for all x except at $x=0$.Given $f(1)\neq 0$

If $f(x.y)=f(x).f(y)$ for all $x,y$ and $f(x)$ is continuous at $x=1$,then show that $f(x)$ is continuous for all x except at $x=0$.Given $f(1)\neq 0$ In the functional equation $f(x.y)=f(x).f(y)$,...

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