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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 $f(x)$,...
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### 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.
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### 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 ...
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### 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!
### 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 ...