6
votes
4answers
83 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 $\mathbb{R}$. ...
1
vote
2answers
70 views

Identifying the exponential function $f(x)=e^x$ from its functional equation

Prove that if $f(x+y)=f(x)f(y)$ for all $x,y$ and $f(x)=1+xg(x)$ where $\lim_{x\to 0}g(x)=1$, then: a) $\exists f'(x)$ $\forall x$ b) $f(x)=e^x$ I would really appreciate your help.
3
votes
0answers
27 views

Convergence of sum of antiderivative and derivative

This question is inspired by this question: Solutions for $ \frac{dy}{dx}=y $?. It makes me wonder if there are any function where the sum of all antiderivative and derivative converges. The ...
9
votes
2answers
97 views

Functions $f$ such that $f(x)+f(-x)=f(x)f(-x)$

I was looking for examples of real valued functions $f$ such that $f(x)+f(-x)=f(x)f(-x)$. Preferably, I'd like them to be continuous, differentiable, etc. Of course, there are the constant functions ...
4
votes
2answers
176 views

Find $f$ such that $\frac{d^2}{dx^2}f(x)=f(\sqrt{x})$

Which non-constant functions $f$ (if any) satisfy $\dfrac{d^2}{dx^2}f(x)=f(\sqrt{x})$ for $x>0$? I suspect there is no $f$ which satisfies the differential equation, but I cannot prove this.
7
votes
1answer
149 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 ...
1
vote
0answers
21 views

Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which ...
7
votes
4answers
169 views

A function satisfying $f(\frac1{x+1})\cdot x=f(x)-1$ and $f(1)=1$?

$f:[0,\infty)\to\mathbb{R}$ is a continuous function which satisfies $f(1)=1$ and: $$f(\frac1{x+1})\cdot x=f(x)-1$$ Does there exist such a function, if they do, are there infinitely many? And is ...
0
votes
0answers
25 views

Can Duhamel's Formula be applied Functional Equations?

Consider an n'th order linear Differential Equation of the form $$y^{[n]} = a_0(x) + a_1(x)y + a_2(x)y' + a_3(x)y'' \ ... \ + a_{n-1}(x)y^{[n-1]}$$ It is well known that to solve this we can create ...
6
votes
3answers
157 views

functions satisfying $f(x)=2f(2x)$

How can I prove every smooth real function f satisfying $f(x)=2f(2x)$ is of the from $f(x)=k/x$ where $k$ is a constant? I have tried this for integers that can be divided by $2^n$, but then I cannot ...
7
votes
2answers
124 views

Continuous functions that satisfies $f(x) + f(1-x) + f(\sqrt{x^2+(1-x)}) = 0$ and $f(\frac12)=0$

$f:[0,1]\rightarrow \mathbb{R}$ is a continuous function which satisfies $$ f(x) + f(1-x) + f\left(\sqrt{x^2+(1-x)}\right) = 0 \text{ and } f\left(\tfrac12\right)=0. $$ Can someone give explicit ...
2
votes
1answer
33 views

A problem about functional equations

We want to find all continuous functions $f:R→R$ that satisfy the equation $f(x^2+1/4)=f(x)$ for all real x. Of course -If I am right- constant functions satisfy the equation mentioned, and as well ...
3
votes
2answers
60 views

Determine the smallest number P

I have here a hard problem, which I couldn't solve. Denote $M$ the set of all functions $f:[0,1]\to\mathbb{R}$ with the following properties: $f(x)\ge0, \forall x$ in $[0,1]$, $f(1)=1$, $f(x+y)\ge ...
0
votes
1answer
127 views

If $f(2x)=2xf'(x)$, then find $f(x)$

If $f(x)$ is Analytic functions on $R$,and such $$2xf'(x)=f(2x)$$ Find all $f(x)$ My idea: let $$f(x)=\sum_{n=0}^{\infty}a_{n}x^n$$ so I can't Thank you
4
votes
2answers
123 views

Find all functions ${\rm f} :{ \mathbb R}_{+}\to{ \mathbb R}_{+}$

Find all functions ${\rm f}:{\mathbb R}_{+} \to {\mathbb R}_{+}$ , such that $\forall\ x,y \in \mathbb R_+$ the equation $$ \left[1 + y{\rm f}\left(x\right)\right]\left[1 - y{\rm f}\left(x + ...
13
votes
4answers
484 views

Find all bijections $\,\,f:[0,1]\rightarrow[0,1]\,$ satisfying $\,\,f\big(2x-f(x)\big)=x$.

A friend of mine gave me the following problem: Find all functions $f:[0,1]\to[0,1]$, which are one-to-one and onto and satisfy the functional relation $$ f\big(2x-f(x)\big)=x, \tag{1} $$ for all ...
8
votes
2answers
268 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 $Q(X)=P(X^2-2) = (P(X))^2-2$ ...
27
votes
2answers
559 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 $$ f(x)+f(x^2)=x,x\in [0,1]. $$ My try: Assume that there is a continuous function with this property. Thus, for any $n\ge 1$ ...
1
vote
1answer
49 views

When can an arbitrary function be put into this form?

A problem I've been trying to address but not been able to get very far with is to devise a method to check whether or not some function $q(x)$ can be written like $$ q(x) = ...
3
votes
1answer
139 views

Functional equations leading to sine and cosine

This question is a possibly harder version of: Find $g'(x)$ at $x=0$. Question. Let $f,g :\mathbb R\to\mathbb R$, such that \begin{align} f(x-y)=f(x)\, g(y)-f(y)\, g(x), \tag{1}\\ g(x-y)=g(x)\, ...
3
votes
2answers
287 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 $f(1/x)-f(x+1)=x$?
17
votes
1answer
341 views

Find $f(x)$ where $ f(x)+f\left(\frac{1-x}x\right)=x$

What function satisfies $ f(x)+f\left(\frac{1-x}x\right)=x$ ?
7
votes
1answer
176 views

A functional relation which is satisfied by $\cos x$ and $\sin x$

Assume that the functions $f,g : \mathbb R\to \mathbb R$ satisfy the relations \begin{align} \left\{ \begin{array}{ll} f(x+y) &=& f(x)f(y)-g(x)g(y), \\ g(x+y) &=& f(x)g(y)+f(y)g(x), ...
19
votes
2answers
582 views

$\cos x\,$ is the only function satisfying $\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y.$

I need to find all continuous functions $f$ which satisfy the functional equation $$ f(x)\,f(y)-f(x+y)=\sin x\,\sin y, $$ for all $x,y\in\mathbb R$. I can prove that ...
1
vote
1answer
50 views

Solving limit equations

Notice the following: all first order differential equations take on the form: $$G(x,f(x),\frac{d}{dx}[f(x)]) = 0 $$ notice that we can replace $\frac{d}{dx}[f(x)]$ with the expression $$ ...
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 $(\alpha_n)_{n=1}^{\infty}$ ...
9
votes
2answers
295 views

How to prove that $f(f(x))=-x$ implies that $f$ is not continuous? [duplicate]

I am trying to prove that: Given an $f:\mathbb{R} \rightarrow \mathbb{R}$, if $f(f(x))=-x$ then $f$ is not continuous? any help? Thank you!
0
votes
0answers
47 views

Integral Functional Equation

If $f(x) = 1 + \int_{x/c}^1 f(t) dt$, find $f(x)$ in terms of $c$. This problem was inspired by trying to find the expected value of the longest increasing subsequence $a_1, a_2, a_3, \cdots, a_n$ ...
0
votes
2answers
111 views

What method is used to find the expression of a function?

Hi everybody I've found some difficulties in this exercise please could you give me help: let $f$ continuous function in $\mathbb R$ $$\forall(x;y)\in\mathbb R, f (x+y) + f(x-y) = 2(f(x)+f(y))$$ 1 - ...
3
votes
1answer
136 views

A Functional Differential Equation

I was having a play with some trig. identities and noticed the following: $\cos{x}=\frac{\sin{2x}}{2\sin{x}}$ Now, $\cos{x} = \frac{d}{dx}\sin{x}$ so I made the following analogous differential ...
0
votes
1answer
147 views

Function $f(x)$ such that $f(x-i)+f(x)=\frac{1}{x^2}$

Help me find a function $f(x)$ such that $$f(x-i)+f(x)=\frac{1}{x^2}$$ where $i$ is the imaginary unit.
3
votes
1answer
34 views

A question on functional equations.

Question: If it is given that $$ e^xf(x) = 2 + \int_0^x\sqrt{1+x^4}\,dx $$ then what is the value of $ \dfrac {d} {dx} \Big(f^{-1}(x)\Big)\Bigg|_{x=2} $ Where I am stuck: Now, since we are to ...
2
votes
1answer
98 views

Find $f(x)$ from $2f'(x)-3f'(1/x)=x$

Find $f(x)$ given that $2f'(x)-3f'(1/x)=x$ Also, is it possible to do this without integration?
1
vote
1answer
69 views

Functional equations very like the Taylor Series

Let $g(x,y)=0$ be a closed curve, that means, any point inside that curve satisfies $g(x,y)<0$ and any point outside that curve satisfies $g(x,y)>0$. Given a point $(a,b)$ outside the curve ...
0
votes
0answers
100 views

integral equations - i need help to expand the function [duplicate]

I have the following integral equation to solve: $$\int_{0}^{2\pi} (\cos^2(x+y)+1/2) \phi (y) dy$$ So, I need to find $\lambda$ where $\lambda$ is the eigenvalues's function. Well, my main goal is ...
-2
votes
2answers
122 views

Integral equation that's cant solve… Need a hand [closed]

Help me solve this integral equation, I'm having some troubles... I need to use the Fredholm method for second kind integral equations. $$\phi(x)= \sin(x)+ \lambda \int_{0}^{\pi}\cos(2x+y)\phi ...
4
votes
3answers
1k views

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

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.
12
votes
2answers
739 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?
7
votes
3answers
276 views

$f(x^2) = 2f(x)$ and $f(x)$ continuous

I ran into a problem recently where I obtained the following constraint on a function. $$f(x^2) = 2f(x) \,\,\,\, \forall x \geq 0$$ and the function $f(x)$ is continuous. Can we conclude that $f(x)$ ...
13
votes
1answer
429 views

Functions $f$ satisfying $ f\circ f(x)=2f(x)-x,\forall x\in\mathbb{R}$.

How to prove that the continuous functions $f$ on $\mathbb{R}$ satisfying $$f\circ f(x)=2f(x)-x,\forall x\in\mathbb{R},$$ are given by $$f(x)=x+a,a\in\mathbb{R}.$$ Any hints are welcome. Thanks.
7
votes
4answers
628 views

If $f(x/2)=f(x)/2$, then $f(x)=f'(0)x$

Let $f:\mathbb R \to \mathbb R$ be differentiable such that $f(x/2)=f(x)/2$ for any $x\in \mathbb R$. How can I prove that $f(x)=f'(0)x$, for any $x\in \mathbb R$? It seems easy, but I don't know why, ...
3
votes
1answer
177 views

Integral Inequality $|f''(x)/f(x)|$

Let $f$ be a $C^2$ function in $[0,1]$ such that $f(0)=f(1)=0$ and $f(x)\neq 0\,\forall x\in(0,1).$ Prove that $$\int_0^1 \left|\frac{f{''}(x)}{f(x)}\right|dx\ge4$$
10
votes
4answers
580 views

The functional equation $f(x)+f(y)+f(z)+f(x+y+z)=f(x+y)+f(y+z)+f(z+x)$

How to find the all continuous functions $f\colon \mathbb{R}\to\mathbb{R}$ such that $f(x)+f(y)+f(z)+f(x+y+z)=f(x+y)+f(y+z)+f(z+x)$
5
votes
6answers
1k views

What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$?

What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$? I'm especially curious if there is both an algebraic and calculus-based derivation of the solution.
2
votes
2answers
170 views

Positive twice differential decreasing function, is it convex?

If g is a positive, twice differentiable function that is decreasing and has limit zero at infinity, does g have to be convex? I am sure, from drawing a graph of a function which starts off as being ...
5
votes
4answers
303 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 ...
4
votes
1answer
446 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: ...
1
vote
0answers
336 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 ...
5
votes
2answers
985 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 $f(x)=ax$?