The name "functional equation" is used for problems where the goal is to find all functions satisfying the given equation (and maybe some other conditions). So in this case, solving the equation means finding all functions fulfilling the equation. (This is different from the more common use of the ...

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21
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1answer
3k 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 ...
17
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3answers
3k 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!
9
votes
3answers
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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) + ...
9
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3answers
2k views

Proving that an additive function $f$ is continuous if it is continuous at a single point

Suppose that $f$ is continuous at $x_0$ and $f$ satisfies $f(x)+f(y)=f(x+y)$. Then how can we prove that $f$ is continuous at $x$ for all $x$? I seems to have problem doing anything with it. Thanks in ...
1
vote
1answer
803 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 $a>0$ ...
8
votes
2answers
786 views

On sort-of-linear functions

Background A function $ f: \mathbb{R}^n \rightarrow \mathbb{R} \ $ is linear if it satisfies $$ (1)\;\; f(x+y) = f(x) + f(y) \ , \ and $$ $$ (2)\;\; f(\alpha x) = \alpha f(x) $$ for all $ x,y \in ...
19
votes
4answers
918 views

thoughts about $f(f(x))=e^x$

I was thinking, inspired by mathlinks, precisely from this post, if there exists a continuous real function $f:\mathbb R\to\mathbb R$ such that $$f(f(x))=e^x.$$ However I have not still been able to ...
97
votes
7answers
4k views

Continuous function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?

I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success: Find a continuous ...
14
votes
2answers
941 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 $c \in \mathbb{R}$ ...
31
votes
8answers
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How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$?

How to get $f(x)$, if we know that $f(f(x))=x^2+x$? Is there an elementary function $f(x)$ that satisfies the equation?
2
votes
2answers
1k views

Graph of discontinuous linear function is dense

$f:\mathbb{R}\rightarrow\mathbb{R}$ is a function such that for all $x,y$ in $\mathbb{R}$, $f(x+y)=f(x)+f(y)$. If $f$ is cont, then of course it has to be linear. But here $f$ is NOT cont. Then show ...
6
votes
3answers
883 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 ...
6
votes
4answers
329 views

3rd iterate of a continuous function equals identity function

If $ f: \mathbb{R} \to \mathbb{R} $ is continuous, and $\forall x \in \mathbb{R} :\;(f \circ f \circ f)(x) = x $, show that $ f(x) = x $. The condition that $f$ is continuous on $\mathbb{R}$ is ...
5
votes
1answer
2k views

Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions…)

My lecturer was talking today (in the context of probability, more specifically Kolmogorov's axioms) about the additive property of functions, namely that: $$f(x+y) = f(x) + f(y)$$ I've been trying ...
63
votes
7answers
2k views

Does a non-trivial solution exist for $f'(x)=f(f(x))$?

Does $f'(x)=f(f(x))$ have any solutions other than $f(x)=0$? I have become convinced that it does (see below), but I don't know of any way to prove this. Is there a nice method for solving this kind ...
4
votes
1answer
270 views

A question concerning on the axiom of choice and Cauchy functional equation

The Cauchy functional equation: $$f(x+y)=f(x)+f(y)$$ has solutions called 'additive functions'. If no conditions are imposed to $f$, there are infinitely many functions that satisfy the equation, ...
6
votes
3answers
698 views

I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ which are continuous and satisfy $f(x+y)=f(x)+f(y)$

I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $f(x+y)=f(x)+f(y)$. I know that there are other questions that are asking the same thing, but I'm trying to figure this out ...
8
votes
3answers
707 views

How to calculate $f(x)$ in $f(f(x)) = e^x$?

How would I calculate the power series of $f(x)$ if $f(f(x)) = e^x$? Is there a faster-converging method than power series for fractional iteration/functional square roots?
4
votes
1answer
112 views

Are Exponential and Trigonometric Functions the Only Non-Trivial Solutions to $F'(x)=F(x+a)$?

Are exponential & trigonometric functions the only non-trivial solutions to $F'(x)=F(x+a)$? $F(x)=0$ would be the trivial solution. Then, for $a=0$ (or $a=2\pi i$), we have $F(x)=e^x$, and ...
4
votes
2answers
2k views

How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?

This is part of a homework assignment for a real analysis course taught out of "Baby Rudin." Just looking for a push in the right direction, not a full-blown solution. We are to suppose that ...
28
votes
7answers
4k views

Find $f(x)$ such that $f(f(x)) = x^2 - 2$

Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$. Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but ...
25
votes
6answers
2k views

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$
25
votes
2answers
644 views

$f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$

A function that satisfies both $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$ for all real $x$ is known to be the identity over $\mathbb Q$, but is it also the identity over $\mathbb R$? If not, can you provide ...
10
votes
1answer
398 views

A unsolved puzzle from Number Theory/ Functional inequalities

The function $g:[0,1]\to[0,1]$ is continuously differentiable and increasing. Also, $g(0)=0$ and $g(1)=1$. Continuity and differentiability of higher orders can be assumed if necessary. The ...
8
votes
1answer
523 views

Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.

I've recently encountered this strangely attractive equation (Riemann's functional equation), along with Riemann's original proof. $$\displaystyle\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) ...
11
votes
1answer
290 views

Is there a real-valued function $f$ such that $f(f(x)) = -x$?

Is there a function $f\colon \mathbb{R} \to\mathbb{R} $ such that $ f(f(x)) = -x$ ?
10
votes
2answers
578 views

Entire functions such that $f(z^{2})=f(z)^{2}$

I'm having trouble solving this one. Could you help me? Characterize the entire functions such that $f(z^{2})=f(z)^{2}$ for all $z\in \mathbb{C}$. Hint: Divide in the cases $f(0)=1$ and $f(0)=0$. ...
7
votes
5answers
1k views

Find polynomials such that $(x-16)p(2x)=16(x-1)p(x)$

Find all polynomials $p(x)$ such that for all $x$, we have $$(x-16)p(2x)=16(x-1)p(x)$$ I tried working out with replacing $x$ by $\frac{x}{2},\frac{x}{4},\cdots$, to have $p(2x) \to p(0)$ but then ...
7
votes
1answer
1k views

Solution for exponential function's functional equation by using a definition of derivative

let $f(0)=1$ and $f'(0)=1$. and $f(x+y)=f(x)f(y)$ for $x,y\in R$. How can I found $f(x)$ by using a definition of derivative?
6
votes
1answer
324 views

Iterated polynomial problem

Polynomial $P$ satisfies $P(n)>n$ for all positive integers $n$. Every positive integer $m$ is a factor of some number of the form $P(1),P(P(1)),P(P(P(1))),\ldots $. Prove that $P(x)=x+1$.
1
vote
6answers
233 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 ...
5
votes
2answers
1k views

How to prove $f(x)=ax$ if $f(x+y)=f(x)+f(y)$ and $f$ is locally integrable

Suppose $f(x)$ is integrable in any bounded interval on $\mathbb R$, and it satisfies the equation $f(x+y)=f(x)+f(y)$ on $\mathbb R$. How to prove $f(x)=ax$?
1
vote
1answer
436 views

Solution of Cauchy functional equation which has an antiderivative

Let $f\colon\mathbb R\to\mathbb R$ be a function such that $$f(x+y)=f(x)+f(y)$$ for any $x,y\in\mathbb R$ i.e., it fulfills Cauchy functional equation. Additionally, suppose that $F'=f$ for some ...
24
votes
3answers
3k views

Prove that this function is bounded

This is an exercise from Problems from the Book by Andreescu and Dospinescu. When it was posted on AoPS a year ago I spent several hours trying to solve it, but to no avail, so I am hoping someone ...
28
votes
2answers
807 views

How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$.

Prove that there is no continuous functions $f:[0,1]\to R$, such that $$ f(x)+f(x^2)=x. $$ My try. Assume that there is a continuous function with this property. Thus, for any $n\ge 1$ and all ...
11
votes
1answer
329 views

Additive functional inequality

The function $f:R_+\to R_+$ is continuously differentiable and increasing. Also, $f(0)=0$ and $f(\infty)=\infty$. Continuity and differentiability of higher orders can be assumed if necessary. ...
16
votes
4answers
1k views

About the addition formula $f(x+y) = f(x)g(y)+f(y)g(x)$

Consider the functional equation $$f(x+y) = f(x)g(y)+f(y)g(x)$$ valid for all complex $x,y$. The only solutions I know for this equation are $f(x)=0$, $f(x)=Cx$, $f(x)=C\sin(x)$ and $f(x)=C\sinh(x)$. ...
11
votes
3answers
335 views

Is Zorn's lemma necessary to show discontinuous $f\colon {\mathbb R} \to {\mathbb R}$ satisfying $f(x+y) = f(x) + f(y)$?

A UC Berkeley prelim exam problem asked whether an additive function $f\colon {\mathbb R} \to {\mathbb R}$, i.e. satisfying $f(x + y) = f(x) + f(y)$ must be continuous. The counterexample involved ...
16
votes
3answers
2k views

The easy(?) part of IMO 2011 Problem 3

Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies $$f(x + y) \leq yf(x) + f(f(x))$$ for all real numbers $x$ and $y$. How can I prove that ...
10
votes
3answers
263 views

Solving $(f(x))^2 = f(\sqrt{2}x)$

I would like to know how to solve this equation : $$f(x)^2 = f(\sqrt{2}x)$$ We assume that $f : \mathbb R \to \mathbb R$ is $\mathcal C^{2}$. The answer should be $f(x)=e^{-x^{2}/2}$, but I don't ...
10
votes
1answer
664 views

Riemann's thinking on symmetrizing the zeta functional equation

In the translated version of Riemann's classic On the Number of Prime Numbers less than a Given Quantity, he quickly derives the zeta functional equation through contour integration essentially as ...
9
votes
3answers
420 views

Evaluating $f(x) f(x/2) f(x/4) f(x/8) \cdots$

Let $f : \mathbb R \to \mathbb R$ be a given function with $\lvert f(x) \rvert \le 1$ and $f(0) = 1$. Is there a nice simplified expression for $$\begin{align}F(x) &= f(x) f(x/2) f(x/4) f(x/8) ...
13
votes
5answers
397 views

If $f:\mathbb{R}\to\mathbb{R}$ is continuous and $f(x)\neq x$ for all $x$, must it be true that $f(f(x))\neq x$ for all $x$?

Let $f: \Bbb R → \Bbb R$ be a continuous function such that $f(x)=x$ has no real solution . Then is it true that $f(f(x))=x$ also has no real solution ?
12
votes
2answers
1k views

Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?

If a Continuous function $f(x)$ satisfies $f(x) = f(2x)$, for all real $x$, then does $f(x)$ necessarily have to be constant function? If so, how do you prove it? If not any counter examples?
11
votes
7answers
2k views

How to find the function $f$ given $f(f(x)) = 2x$?

I was wondered how to find the function in this equality: $f(f(x))=2x$. Also $f$ is continuous. I don't need the answer, how to find it is more important.
9
votes
3answers
518 views

a continuous function satisfying $f(f(f(x)))=-x$ other than $f(x)=-x$

My question is about existence of a non-trivial solution of the functional equation $f(f(f(x)))=-x$ where $f$ is a continuous function defined on $\mathbb{R}$. Also, what about the general one ...
7
votes
1answer
321 views

Find all differentiable functions $f$ such that $f\circ f=f$

I want to find all differentiable functions $f:\mathbb R \to \mathbb R$ such that $f\circ f=f$, My attempt since $f$ is differentiable, $f'(f(x))f'(x)=f'(x)$ Now if $f'(x)\neq0$($f'=0$ means ...
5
votes
3answers
422 views

Inverse function of $y=W(e^{ax+b})-W(e^{cx+d})+zx$

I have a simple question for which I am looking for a closed form expression (If there exits one). In other words, given: $$y=W(e^{ax+b})-W(e^{cx+d})+zx$$ where $W$ is the Lambert $W$ function and ...
7
votes
4answers
166 views

$f:\Bbb Z\to\Bbb Z$ such that $f(f(n)-2n)=2f(n)+n$ for all $n\in\Bbb Z$

Does there exists a function $f:\Bbb Z\to\Bbb Z$ such that $$f(f(n)-2n)=2f(n)+n$$ for all $n\in\Bbb Z$
4
votes
3answers
309 views

Strategies to find the set of functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy a given functional equation

My question is as follows: What methods can be used to find the set of functions $f:\mathbb{R}\to\mathbb{R}$ satisfying a certain functional equation. An example of a case where this applies is the ...