1
vote
2answers
60 views

Find all functions $ f:\Bbb{R}\to\Bbb{R}$ with the intermediate value property such that $\exists 1\leq n\in\Bbb{Z}, f^n(x)=-x$

I got this problem: Prove that the only function $ f:\Bbb{R}\to\Bbb{R}$ with the intermediate value property such that $\exists 1\leq n\in\Bbb{Z}, \forall x\in\Bbb{R}, f^n(x)=-x$ where $f^n =f\circ ...
3
votes
1answer
36 views

Find examples of continuous functions $f:[0,1]\to[0,1]$ that satisfy $\forall x\in[0,1], f(f(x))=f(x)$ other than $f(x)=x$

I got this problem: Find examples of continuous functions $f:[0,1]\to[0,1]$ that satisfy $\forall x\in[0,1], f(f(x))=f(x)$ other than $f(x)=x$. I proved that $f([0,1])=\{x\in[0,1]|f(x)=x\}$, But I ...
8
votes
4answers
154 views

Prove that there is no function $f:\Bbb{R}\to\Bbb{R}$ with $f(0)>0$ such that $\forall x,y\in\Bbb{R}, f(x+y)\geq f(x)+y f(f(x))$

I got this problem: Prove that there is no function $f:\Bbb{R}\to\Bbb{R}$ with $f(0)>0$ such that $\forall x,y\in\Bbb{R}, f(x+y)\geq f(x)+y f(f(x))$. (Hint: the solution involves limits at ...
6
votes
4answers
87 views

Continuity of function where $f(x+y) = f(x)f(y) ~~\forall x, y \in \mathbb{R}$.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function which satisfies $f(x+y) = f(x)f(y) ~~\forall x, y \in \mathbb{R}$ is continuous at $x=0$, then it is continuous at every point of $\mathbb{R}$. ...
0
votes
0answers
23 views

Existence of a function satisfying all the given infima

Given a function $f : \mathbb{R} \to \mathbb{R}$, we can compute its infimum over $A$ for all the Borel measurable $A \subset \mathbb{R}$. I am wondering when we can deduce in the other direction, ...
0
votes
0answers
82 views

Functional equation relating to normal numbers

My coauthor and I have run into the following problem in a research project involving normal numbers. We suspect that the following question may be resolved using standard techniques in analysis. We ...
0
votes
2answers
45 views

What functions satisfy the condition $f(x,y)=g(x)$?

Are there any functions $f(x,y)$ and $g(x)$ that satisfy (1) $f(x,y)=g(x)$ for all $x \in \mathbb{R}$ and $y\in \mathbb{R}$ (2) $f(x,y)$ is not constant in $y$ for each $x$ (i.e. for each $x$ there ...
3
votes
0answers
62 views

Continuous functions such that $f(x)+f(x^2)=x$ [duplicate]

Find all continuous $f :[0,1]\to \mathbb R$ such that $\forall x\in [0,1], f(x)+f(x^2)=x$ I suspect there are none. I made little progress so far, but it's worth noticing that $f(0)=0$ and ...
7
votes
1answer
152 views

Functional equation and fixed points

Let $f$ be strictly increasing and such that $f(x)+f^{-1}(x)+1=e^x$. Is it true that $f$ has at most one fixed point? I am told the answer is yes, but I am having trouble proving it. It's obvious ...
11
votes
1answer
317 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. ...
8
votes
4answers
214 views

Real-analytic $f(z)=f(\sqrt z) + f(-\sqrt z)$?

Are there nonconstant real-analytic functions $f(z)$ such that $$ f(z)=f(\sqrt z) + f(-\sqrt z)$$ is satisfied near the real line ? Also can such functions be entire ? And/Or can they be periodic ...
3
votes
1answer
123 views

$f,g$ such that $\int fg = \int f \int g$

Suppose $f,g$ are real valued on $\mathbb{R}$ (and no further restrictions apart from the obvious requirement that the integrals exist), then when does $\displaystyle\int f(x)g(x)\,dx = \int f(x) \, ...
10
votes
4answers
173 views

If $f(3x)=f(x)$ and $f$ is continuous, show that $f(x)$ is a constant function.

If $f(x)$ is a continuous function such that $f(3x)=f(x)$ and the domain of $f$ is all non-negative real numbers. Prove that $f$ is a constant function. What I did: ...
5
votes
3answers
116 views

Functional equation $f(x)=f(\sqrt{x})$

If we take an equation $f(x)=f(\sqrt{x})$ defined for positive $x$ then it is quite easy to see that it is constant; $f(x)=f(0)$ if continuous at zero. My question is: What would happen if we take ...
1
vote
2answers
72 views

Functional equation (is solution unique)

Let $f:[0,1]\rightarrow \mathbb{R}$ - cont. diff. function. Is it true that equation $\cos{t}f(\sin{t})+\sin{t}f(\cos{t})=1, t \in [0,\pi/2]$ has only solution $f(x)=\sqrt{1-x^2}$? How can we prove ...
4
votes
1answer
63 views

submodular-like functions on $\mathbb{R}$

The definition of submodularity led me to define a type of function on numbers (I suppose ordered rings, but consider the reals for now) as $f(x) + f(y) \ge f(x+y) + f(xy)$. Such functions exist: for ...
7
votes
0answers
66 views

Find all pairs of functions $(f,g)$, $\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$

Find all pairs of functions $(f,g)$ : $\mathbb{R} \to \mathbb{R}, g : \mathbb{R} \to \mathbb{R}$ satisfying : $$\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$$ I am really ...
6
votes
0answers
93 views

How to prove subadditive function?

Let $f: [0, \infty) \to \Bbb R$ be a function satisfying the following conditions: (1) For any $x,y \geq 0, f(x+y) \geq f(x) + f(y)$. (2) For any $x \in [0,2], f(x) \geq x^2 - x$. Prove that, for ...
1
vote
0answers
58 views

Another functional equation

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
2
votes
1answer
35 views

About the space $u \in C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})$

I am reading Taylor's Partial differential equations III (nonlinear equations) (Section 1 of Chapter 16, Theorem 1.2), and Sogge's Lectures on Non-linear wave equatuions. I notice that in the energy ...
-1
votes
1answer
72 views

Does there exist a continuous function $f\colon \Bbb R\rightarrow \Bbb R$ such that $f(f(x))=-x$ for all $x\in\Bbb R$? [duplicate]

Does there exist a continuous function $f\colon \Bbb R\rightarrow \Bbb R$ such that $f(f(x))=-x$ for all $x\in\Bbb R$?
1
vote
0answers
45 views

Can the Euler-Lagrange equations be derived from a variation over a time of order $dt$ rather than $t$?

In the calculus of variations, the solution of the Euler-Lagrange equations gives those functions for which a given functional is stationary. Now all derivations I've come across up to now, carry out ...
4
votes
1answer
62 views

Find all function satisfying a condition with $\min$ and $\max$

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$ \forall (x,y)\in\mathbb{R}^2,x\ne y,\quad \min (f(x),f(y)) \leq \frac{f(x)-f(y)}{x-y} \leq \max(f(x),f(y)) $$ I have started with ...
0
votes
4answers
103 views

Functional equations

Let $f:\mathbb{R}\to \mathbb{R}$ is a function such that for all real $x$ and $y$, $f(x+y)= f(x) + f(y)$ and $f(xy)= f(x)f(y)$, then prove that $f$ must be one of the two following functions: ...
7
votes
2answers
79 views

Non trivial solutions of $g\circ f-f\circ g=g\circ f\circ g$

While thinking of perfect numbers, I came across the functional equation $g\circ f-f\circ g=g\circ f\circ g$ where the unknowns $f$ and $g$ are functions from $\mathbb{R}$ to itself. I only know one ...
8
votes
2answers
274 views

How to find all polynomials P(x) such that $P(x^2-2)=P(x)^2 -2$?

I am trying the fallowing exercise : Solve $P(X^2 -2)=P(X)^2 -2$ with P a monic polynomial (non-constant) My attempt : Let P satisfying $P(X^2-2) = (P(X))^2-2$ Then $Q(X)=P(X^2-2) = (P(X))^2-2$ ...
7
votes
1answer
228 views

Find all continuous functions $ f:\mathbb{R} \rightarrow \mathbb{R}$ such that

Find all continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x)^2 - 2\Bigl\lfloor\,2\,\bigl|f(2x)\bigr|\,\Bigr\rfloor f(x) + \Bigl\lfloor{4f(2x)^2-1}\Bigr\rfloor = 0\,.$$ Here ...
3
votes
2answers
314 views

Find a solution for f(1/x)+f(1+x)=x

Title says all. If f is an analytic function on the real line, and $f(1/x)+f(x+1)=x$, what, if any, is a possible solution for $f(x)$? Additionally, what are any solutions for $f(1/x)-f(x+1)=x$?
2
votes
1answer
49 views

Existence of a function with a changing period

$f,\alpha$ are continuous $\mathbb{R}\to\mathbb{R}$ functions satisfying: $$f\big(x+\alpha(x)\big)=f(x)$$ If $f$ is non-constant, must $\alpha$ be constant? My idea was to use the fact ...
4
votes
5answers
153 views

Is there an everywhere-defined function that satisfies $f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}$

Is there a function $f:\mathbb{R}\to\mathbb{R}$ which is differentiable and satisfies the following: (1) $f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}$ (2) $f'(0)=1$ (1) is the functional equation for ...
3
votes
1answer
157 views

Real analytic $f$ satisfying $f(2x)=f(x-1/4)+f(x+1/4)$ on $(-1/2,1/2)$

What are the real analytic functions $f\colon (-\frac{1}{2},\frac{1}{2}) \to \mathbb{C}$ that satisfy the following functional equation, and how are they derived? ...
8
votes
2answers
183 views

Continuous $f$ satisfying $f(2x)=f(x-1/4)+f(x+1/4)$ on $(-1/2,1/2)$

What are the continuous functions $f\colon (-\frac{1}{2},\frac{1}{2}) \to \mathbb{C}$ that satisfy the following functional equation, and how are they derived? ...
4
votes
3answers
90 views

Retrieving the Taylor series for $\log$ from its functional equation

Consider the unique continuous function $\mathbb{R}^+\to\mathbb{R}$ such that: $$f(xy)=f(x)+f(y),\qquad f(e)=1$$ where $\displaystyle e=1+\sum_{n=1}^{\infty} \frac{1}{n!}$. Assuming $f$ has a ...
1
vote
2answers
72 views

functional equations with restricted domain

How can I find all discontious solutions of functional equation $f(xy)=f(x)f(y)$ on $[0,1]$. Similar question is to find all solutions of the equation $f(x+y)=f(x)+f(y)$ on $[0,\infty)$. Can we still ...
6
votes
1answer
245 views

Find all continuous functions $f$ : $\mathbb{R} \rightarrow \mathbb{R}$ satisfied that: $f(xy)+f(x+y)=f(xy+x)+f(y)\quad\forall x,y \in \mathbb{R}$

Find all continuous functions $f$ : $\mathbb{R} \rightarrow \mathbb{R}$ satisfied that: $$f(xy)+f(x+y)=f(xy+x)+f(y)$$ $\forall x,y \in \mathbb{R}$ I have tried that : $P(y;x)-P(x;y)$: ...
2
votes
3answers
225 views

Really nice functional equation with second partial deratives.

Show that $f:\mathbb{R}^2\to\mathbb{R}$, $f \in C^{2}$ satisfies the equation $$\frac{\partial^2 f}{\partial x^2} - \frac{\partial^2 f}{\partial y^2} = 0$$ for all points $(x,y) \in \mathbb{R}^2$ if ...
0
votes
2answers
72 views

Is this $f$ a linear function?

My question is related to this as I posted earlier. But this time, we drop certain conditions: Suppose $f:[a,b]\to\mathbb{R}$ be continuous and there exists a sequence $(\alpha_n)_{n=1}^{\infty}$ ...
3
votes
1answer
67 views

If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ and $f'(0)$ exists, then $f$ is differentiable on $\mathbb R$

If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ and $f'(0)$ exists, then $f$ is differentiable on $\mathbb R$. I know how to show it is continuous but no clue how to show ...
14
votes
1answer
336 views

Condition for an additive function to be continuous

The problem below is Problem 7 from this year's Miklos Schweitzer competition (contest ended Nov 4th). Suppose that $f: \Bbb{R} \to \Bbb{R}$ is an additive function (that is $f(x+y) = f(x)+f(y)$ ...
3
votes
2answers
164 views

Do there exist functions such that $f(f(x)) = -x$? [duplicate]

I am wondering about this. A well-known class of functions are the "involutive functions" or "involutions", which have that $f(f(x)) = x$, or, equivalently, $f(x) = f^{-1}(x)$ (with $f$ bijective). ...
10
votes
2answers
263 views

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$, $f(x)f(yf(x))=f(x+y)$

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$$$f(x)f(yf(x))=f(x+y)$$ A start: set y=0 to get $f(x)f(0)=f(x)$. So $f(0)=1$ unless $f$ is identically zero.
3
votes
1answer
240 views

How to identify this power series as $k\sin(k/x)$?

In this question, a functional equation is solved for functions with a power series. We find a recursive formula: (copied from the answer by user achille hui) \begin{align} ( 2^1 - 3 ) a_2 &= 0\\ ...
6
votes
2answers
430 views

Real Analysis Proofs: Additive Functions

I'm new here and could really use some help please: Let $f$ be an additive function. So for all $x,y \in \mathbb{R}$, $f(x+y) = f(x)+f(y)$. Prove that if there are $M>0$ and $a>0$ such that ...
3
votes
2answers
145 views

Alternatives to percentage for measuring relative difference?

If the value of something changes from $\;a\;$ to $\;b\;$, their relative difference can be expressed as a percentage: $$ \newcommand{\upto}{\mathop\nearrow} (D0) \;\;\; a \upto b \;=\; (b-a)/a ...
2
votes
1answer
57 views

Find all continous functions satistying $ f(x) = f(x^{2} + \frac{x}{3} + \frac{1}{9})$ for all $x \in \mathbb{R}$

The problem I am trying to solve now is to find all continous functions satistying $f(x) = f(x^{2} + \frac{x}{3} + \frac{1}{9})$ for all $x \in \mathbb{R}$ It is the first time for me to face this ...
1
vote
1answer
110 views

$f(x)=f(x^2+ 1/4)$ , $f$ is continuous from $\mathbb{R}$ to $\mathbb{R}$

Find all continous functions $f\colon\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(x)=f(x^2+ 1/4)$ What I've tried so far: suppose that $f$ is one-one thus $x=x^2+1/4$ ... $x=1/2$ then ...
1
vote
0answers
44 views

Does one have to check some additional hypothesis to apply the implicit function theorem for infinite dimensional spaces?

Let $L : B_1 \rightarrow B_1$ be an isomorphism of Banach spaces (i.e., L is a bijective bounded linear operator) and $A: B_1 \rightarrow B_2$ a non linear operator. Consider the following equation ...
0
votes
3answers
76 views

Solve this functional equation: $h(-s)=a-h(s)$

Let $h$ be an analytic function. My question is : Solve this functional equation: $$h(-s)=a-h(s)$$ holds true for all $s∈ℂ$. Here, $a∈ℂ$, $a≠0$.
7
votes
2answers
216 views

Find all real polynomials $p(x)$ that satisfy $\sin( p(x) ) = p( \sin(x) )$

Find all real polynomials $p(x)$ that satisfy $\sin( p(x) ) = p( \sin(x) )$ and prove they are indeed all. Is there an easy way to prove this?
6
votes
0answers
216 views

Differentiable functions satisfying $f'(f(x))=f(f'(x))$

I am wondering whether or not there is a reasonable characterization of differentiable functions $f: \mathbb{R}\to \mathbb{R}$ such that $f'(f(x))=f(f'(x))$ for each $x\in\mathbb{R}$. (Or, if you like ...