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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1answer
775 views

I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive. [duplicate]

Possible Duplicate: Proving that an additive function $f$ is continuous if it is continuous at a single point Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions…) ...
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1answer
298 views

solve functional equation: $[f(x)]^2-[f(y)]^2$=$f(x+y)f(x-y)$

i am trying to solve following problems and please guys help me suppose that,there is given following equation $[f(x)]^2-[f(y)]^2$=$f(x+y) \cdot f(x-y)$ there was said that,it requires some knowledge ...
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0answers
115 views

solution of d’Alembert’s equation.

i know that equation for d’Alembert’s equation. is looking so $g(x+y)+g(x-y)=2*g(x)*g(y)$ so am trying to find actual solution for this equation,first i took $x=y=0$ and i got $2*g(0)$=$2*g(0)^2$ ...
2
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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 ...
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3answers
218 views

What was this theorem called

Back at the university we have proven (lot of work) that if $$S(X)C(Y)+C(X)S(Y) = S(X+Y)$$ and $$C(X)C(Y)-S(X)S(Y) = C(X+Y)$$ then $S(X)$ is $\sin(x)$ and $C(X)$ is $\cos(x)$ (or constant $0$, meh). ...
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1answer
172 views

Analytic solution of a certain functional equation

Here's my question. Let $b_2$, ..., $b_d \in \mathbb{C}$ ($d$ is an integer greater than 2), and consider the functional equation $$V(z^d)=dz^{d-1} V(z)+(b_2 z^{d-2} + b_3 z^{d-3}+\ldots + b_d)$$ ...
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2answers
142 views

Find $f$ such that $\{f(x+y)\}=\{f(x)\}+\{f(y)\}$

Find all continuous function such that $\{f(x+y)\}=\{f(x)\}+\{f(y)\}$ for all $x, y\in\mathbb{R}$. Denote $\{x\}=x-[x]$ in which $[x]$ is the largest integer number does not exceed x.
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2answers
245 views

Recurrence relations on a continuous domain

While attempting to read Shannon's paper I came across the following (p. 3): suppose $N\colon \mathbb{R} \to \mathbb{R}$ is a function, which for some fixed (given) set of values $t_1, t_2, \dots, ...
5
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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 ...
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4answers
318 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 ...
2
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0answers
71 views

Given that $f(x^2)=f(x)^2$ and $f(x+1)=f(x)+1$, try to find $f$ [duplicate]

Possible Duplicate: $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$ Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that for all reals $x$ $f(x^2)=f(x)^2$ and $f(x+1)=f(x)+1$ Can we show that ...
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1answer
41 views

In which cases iterative equations can be reduced to finite-difference equations?

In which cases iterative equations can be reduced to finite-difference equations and when they can't?
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3answers
660 views

$f(x^2) = xf(x)$ implies that $ f(x) = mx$?

Suppose a function $f : \mathbb{R} \to \mathbb{R} $ satisfies the relation $$f(x^2) = xf(x) \ \ \forall x$$ Does this imply $f$ must be a straight line, $f(x) = mx$? If so, why? If not, are there ...
2
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0answers
415 views

Finding inverse of function without knowing function?

This question has a programming application, but I thought it would be more appropriate (and educational) to ask here first and get some basic understanding. So my use-case will refer to some ...
13
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3answers
758 views

very elementary proof of Maxwell's theorem

Maxwell's theorem (after James Clerk Maxwell) says that if a function $f(x_1,\ldots,x_n)$ of $n$ real variables is a product $f_1(x_1)\cdots f_n(x_n)$ and is rotation-invariant in the sense that the ...
11
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1answer
281 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$ ?
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0answers
123 views

Could anyone derive a formula for this?

Edited: I want to get a sentiment score of various sentences and I've tried coming up with an equation that could satisfy the conditions that are inherent to each sentence (It's estimated mood as ...
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2answers
82 views

General aggregation functions

Is there a way to find all or some functions which "aggregate" numbers and are non-isomorphic to addition. I mean functions which are commutative and associative: $f(x,y)=f(y,x)$ ...
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1answer
587 views

Limit of a function satisfying an inequality

If $f(x)+f(y)\leq f(x+y)$ and $f:\mathbb{R}\to\mathbb{R}$, then can we find $\lim_{x\to 0} \frac {f(x)}{x}$? I am not sure whether the question is correct.Thank you.(I tried this idea: ...
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0answers
390 views

functional derivative of an integral of the function itself

I have the following $$ \frac{d}{dn(x)} \int_{x \in \cal{R}^3} {n(x) dx} $$ I know that this additional relationship holds $$ \int_{x \in \cal{R}^3}{n(x) dx} = N $$ where N is a constant. My ...
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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 $f(x)=ax$?
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2answers
592 views

Which trigonometric identities involve trigonometric functions?

Once upon a time, when Wikipedia was only three-and-a-half years old and most people didn't know what it was, the article titled functional equation gave the identity $$ \sin^2\theta+\cos^2\theta = 1 ...
5
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1answer
947 views

Examples of functions where $f(ab)=f(a)+f(b)$

What are some examples of continuous (on a certain interval) real or complex functions where $f(ab)=f(a)+f(b)$ (like $\ln x$?)
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0answers
248 views

A function $f(x)$ such that $f(f(x))=\ln(x)$

How to find a continuous function $f(x)$ for $x > 12$, such that $f(f(x))=\ln(x)$? Preferably analytic too.
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1answer
112 views

Bijective functions $f(n)=f(f(n-1))=f^n(1)$

How can I find a function (or more) which satisfy $f(n)=f(f(n-1))=f^n(1)$, defined for positive integers n, such that it is a bijection between the positive integers? And which satisfy i) For every ...
2
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1answer
457 views

About finding the function such that $f(xy)=f(x)f(y)-f(x+y)+1$

Define a function $f\colon\mathbb{R}\to\mathbb{R}$ which satisfies $$f(xy)=f(x)f(y)-f(x+y)+1$$ for all $x,y\in\mathbb Q$. With a supp condition $f(1)=2$. (I didn't notice that.) How to show that ...
4
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3answers
251 views

Solving the equation $f(x+t)=f(x)+f(t)+2\sqrt{f(x)}\sqrt{f(t)}$

I am trying to solve the equation $f(x+t)=f(x)+f(t)+2\sqrt{f(x)}\sqrt{f(t)}$ - as in find a function that satisfies this equation. I notice that the RHS is $({\sqrt{f(x)}+\sqrt{f(t)}})^2$ but I am ...
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1answer
3k views

What does it mean for a functional equation to have a unique solution?

It is my thinking that unique conventionally means special or one of its kind. But in the context of solving functional equations*, I am confused what it means to have a unique solution... *e.g. Find ...
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2answers
237 views

Is there a continuous {permutation,duplicate,translation,stretch}-invariant function on ordered sets of vectors that returns a vector?

Is there any example of such a function $f$, preferably one defined on all $V^n$ and all positive integers $n$ where $V$ is some vector space? It must satisfy the following: $f(T) = f(\sigma(T))$ ...
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3answers
286 views

What is the solution of $f(x)\cdot f(-x) = 1$

What is the general solution of the equation? $$f(x) \cdot f(-x) = 1$$ I know that $f(x) = A^{k \cdot x}$ is a solution, and I am feeling this is the general solution, but I don't have any proof. ...
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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
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5answers
700 views

$ f(x)=f(x+1) $ and $ f(-1/x)=f(x)$

Is there any function $f$ which would satisfy $f(x)=f(x+1)$ and $f(-1/x)=f(x)$ for every $x$ or at least positive $x$? For the widest possible domains of $x$? If I could turn this functional equation ...
7
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1answer
424 views

Entire function with $f(z)=\sin(f(z))$ must be constant?

I'm trying to show why an entire function with the property $f(z)= \sin(f(z))$ everywhere must be constant. Is it sufficient to say that when taking the derivatives, we will get $f'(z)=f'(z) ...
2
votes
1answer
248 views

How to solve polynomial functional equation $P(x,y)=P(\frac{x-y}{2},\frac{y-x}{2})$?

Given $P(x,y)$ which is a polynomial function, satisfying $P(x,y)=\displaystyle P(\frac{x-y}{2},\frac{y-x}{2})$. Then why should $P(x,y)$ be $\displaystyle\sum^n_{i=0}a_i(x-y)^i$? Is it unique?
3
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0answers
148 views

Analytic function that provide $f^2(z)=z$

I am trying to solve this problem: Does there exist a function $f(z)$, that is analytic at $E=\{x+iy :x>y\}$ and provides $f^2(z)=z$ for every $z \in \mathbb C$. I have seen a solution that ...
2
votes
1answer
276 views

$f(x+1)=f(x)+1$ and $f(g(x))=g(f(x))$

Let $g_1(x)=x+1$ and $g_2(x)=x^2$ be two real functions. Then it is known that whenever $f$ commutes with $g_1$ and $g_2$, $f$ is the identity function. But in this example we choosed 2 particular ...
7
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7answers
378 views

Function satisfying $x = f(f(x))$ and $x \not= f(x)$

Is there a function that would satisfy the following conditions?: $\forall x \in X, x = f(f(x))$ and $x \not= f(x)$, where the set $X$ is the set of all triplets $(x_1,x_2,x_3)$ with $x_i \in ...
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1answer
227 views

functional equation uniqueness $f(x^{2})= f(x)^{2} $ in $\mathbb{C}$ and $f(x)+f''(x) = 0 $ in $\mathbb{C}$

This is an exercise from a book I tried: One would like to find all holomorphic equations that satisfy:$$i) \ f(z)+f''(z) = 0 \text{ in } \mathbb{C} $$$$ii)\ f(z^{2})=f(z)^{2} \text{ in } ...
5
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2answers
398 views

Is there a nontrivial solution to $f(f(f(x)))=-8x$?

Let $f:\mathbb R\to\mathbb R$ be a continuous function such that $f(f(f(x)))=-8x$. Must we have $f(x)=-2x$? I can prove this if I assume that $f$ is continuously differentiable everywhere, but is that ...
19
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6answers
1k views

$f\colon\mathbb{R}\to\mathbb{R}$ such that $f(x)+f(f(x))=x^2$ for all $x$?

A friend came up with this problem, and we and a few others tried to solve it. It turned out to be really hard, so one of us asked his professor. I came with him, and it took me, him and the ...
6
votes
3answers
361 views

Solve $f(f(n))=n!$

What am I doing wrong here: ( n!=factorial ) Find $f(n)$ such that $f(f(n))=n!$ $$f(f(f(n)))=f(n)!=f(n!).$$ So $f(n)=n!$ is a solution, but it does not satisfy the original equation except for ...
6
votes
1answer
150 views

If $f\circ f$ is smooth, is the monotonic function $f$ smooth?

Let $f:\mathbb R\to\mathbb R$ be continuous and monotonic and assume that $f\circ f$ is analytic. Is $f$ necessarily continuously differentiable/smooth/analytic? My question arose from this: Inspired ...
3
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2answers
771 views

Trying to solve the equation involving floor function

I am trying to solve the following equation: Floor[logx+1]+x=11 Where Floor function returns the greatest integer smaller than the value in bracket. e.g. Floor[3.3] = 3 And the logarithm is to the ...
1
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1answer
84 views

Is this a functional equation of a line?

Let $f$ be a real-valued function satisfying the functional equation $$f(x)=f(x+y)+f(x+z)-f(x+y+z)$$ for all $x,y,z\in\mathbb{R}$. Is it true that $f$ must be the equation of a line, with no ...
9
votes
3answers
509 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 ...
8
votes
3answers
655 views

Help remembering a Putnam Problem

I recall that there was a Putnam problem which went something like this: Find all real functions satisfying $$f(s^2+f(t)) = t+f(s)^2$$ for all $s,t \in \mathbb{R}$. There was a cool trick to ...
1
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3answers
262 views

Strange functional equation ( hyperfunctions? )

Can we solve this strange functional equation? $$ f(x+i\epsilon)-f(x-i\epsilon) = g(x) $$ I believe that the solution is the Hilbert (finite part) transform of the function g(x) however I do not ...
0
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1answer
39 views

designing an equation that compares two values and returns a probability

Given two values, I'm trying to come up with a formula that will return 50% if both values are equal, 25% if the first value is half the second, 75% if the second is half the first. In other words: ...
6
votes
1answer
257 views

$ f(x+f(x+y))=f(x-y)+f(x)^2 \quad \forall x,y\in \mathbb R$

We have to find all functions $f\colon \mathbb R\to\mathbb R$ such that f: $$\forall x,y\in \mathbb R \quad f(x+f(x+y))=f(x-y)+f(x)^2.$$ Could somebody help me solve this problem? Thank you.
2
votes
3answers
343 views

Finding an $f(x)$ that satisfies $f(f(x)) = 4 - 3x$

I need to find $f(f(x)) = 4 - 3x$ In other examples, such as $f(2)$, I can see that the result equates to $-2$ or $f(x^2)$ becomes $-3x^2 + 4$. Do I really just substitute $f(x)$ for $x$ and ...