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 ...
12
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3answers
1k views
Is there a name for function with the exponential property $f(x+y)=f(x) \times f(y)$?
I was wondering if there is a name for a function that satisfies the conditions
$f:\mathbb{R} \to \mathbb{R}$ and $f(x+y)=f(x) \times f(y)$?
Thanks and regards!
1
vote
1answer
379 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
2k views
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) + ...
6
votes
3answers
517 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 ...
5
votes
3answers
747 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 ...
5
votes
2answers
323 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 ...
3
votes
1answer
409 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 ...
14
votes
2answers
417 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 $c \in \mathbb{R}$ ...
7
votes
3answers
456 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?
24
votes
2answers
575 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
2answers
450 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
799 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 ...
6
votes
1answer
222 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 $P(x)=x+1$.
5
votes
1answer
439 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?
1
vote
2answers
281 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 ...
14
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 ...
9
votes
3answers
421 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 ...
11
votes
8answers
927 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.
5
votes
3answers
273 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
494 views
Solving the functional equation $f(x+1) - f(x-1) = g(x)$
Given a function $g(x)$, is it possible to find a function $f(x)$ that satisfies
$$ f(x+1) - f(x-1) = g(x) $$
5
votes
4answers
245 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
2answers
183 views
$f(z_1 z_2) = f(z_1) f(z_2)$ for $z_1,z_2\in \mathbb{C}$ then $f(z) = z^k$ for some $k$
Same as my previous question except domain is complex. I tried assuming that the function was analytic, so for $z_1=z_2=z$ , $f(z^2) = f(z)^2$ $$\sum_{n=0}^\infty a_n z^{2n}=\left(\sum_{n=0}^\infty ...
3
votes
1answer
142 views
A question concerning on the axiom of choice and Cauchy functional equation
The Cauchy functional equation:
$$f(x+y)=f(x)+f(y)$$
has solutions called 'additive functions'. If no conditions are imposed to $f$, there are infinitely many functions that satisfy the equation, ...
1
vote
2answers
51 views
Finding Value, Related To Functional Equation
$f(x)$ is continuous for $\forall x \in R$ and $f(2x)-f(x)=x^{3}$
(1) $f(x)+f(-x)$ is constant ?
(2) $f(0)=0$ ?
I don't know how to use the continuity.
especially for $f(0)=0$ ?
0
votes
2answers
53 views
What's the solution of the functional equation
I need help with this:
"Find all functions $f$, $g : \mathbb{Z} \rightarrow \mathbb{Z}$, with $g$ injective and such that:
$$f(g(x)+y) = g(f(x)+y), \mbox{ for all } x, y \in \mathbb{Z}.$$
12
votes
1answer
171 views
Functions $f$ satisfying $ f\circ f(x)=2f(x)-x,\forall x\in\mathbb{R}$.
How to prove that the continuous functions $f$ on $\mathbb{R}$ satisfying
$$f\circ f(x)=2f(x)-x,\forall x\in\mathbb{R},$$
are given by
$$f(x)=x+a,a\in\mathbb{R}.$$
Any hints are welcome. Thanks.
12
votes
4answers
584 views
Solving for the implicit function $f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$ and $f(1)=1$
How can I find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(1)=1$ and
$$f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$$
for all real numbers $x$ and $y$ with $y\neq0$?
PS. This is ...
18
votes
6answers
690 views
Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$
Find all polynomials $P$ such that
$P(x^2+1)=P(x)^2+1$
13
votes
3answers
611 views
Continuous function satisfying $f^{k}(x)=f(x^k)$
How does one set out to find all continuous functions $f:\mathbb{R} \to \mathbb{R}$ which satisfy $f^{k}(x)=f(x^k)$ , where $k \in \mathbb{N}$?
Motivation: Is $\sin(n^k) ≠ (\sin n)^k$ in general?
18
votes
3answers
3k views
Prove that this function is bounded
This is an exercise from Problems from the Book by Andreescu and Dospinescu. When it was posted on AoPS a year ago I spent several hours trying to solve it, but to no avail, so I am hoping someone ...
14
votes
4answers
451 views
Solving (quadratic) equations of iterated functions, such as $f(f(x))=f(x)+x$
In this thread, the question was to find a $f: \mathbb{R} \to \mathbb{R}$ such that
$$f(f(x)) = f(x) + x$$
(which was revealed in the comments to be solved by $f(x) = \varphi x$ where $\varphi$ is ...
9
votes
1answer
207 views
Is there a real-valued function $f$ such that $f(f(x)) = -x$?
Is there a function $f\colon \mathbb{R} \to\mathbb{R} $ such that $ f(f(x)) = -x$ ?
8
votes
1answer
272 views
Riemann's thinking on symmetrizing the zeta functional equation
In the translated version of Riemann's classic On the Number of Prime Numbers less than a Given Quantity, he quickly derives the zeta functional equation through contour integration essentially as
...
6
votes
3answers
156 views
$f(x^2) = 2f(x)$ and $f(x)$ continuous
I ran into a problem recently where I obtained the following constraint on a function.
$$f(x^2) = 2f(x) \,\,\,\, \forall x \geq 0$$
and the function $f(x)$ is continuous. Can we conclude that $f(x)$ ...
6
votes
3answers
146 views
Is there a non-constant function $f:\mathbb{R}^2 \to \mathbb{Z}/2\mathbb{Z}$ that sums to 0 on corners of squares?
A problem in the 2009 Putnam asks about functions $f:\mathbb{R}^2 \to \mathbb{R}$ such that whenever $A,B,C,D$ are corners of some square we have $f(A)+f(B)+f(C)+f(D)=0$. Without spoiling the problem ...
12
votes
4answers
404 views
thoughts about $f(f(x))=e^x$
I was thinking, inspired by mathlinks, precisely from this post, if there exists a continuous real function $f:\mathbb R\to\mathbb R$ such that $$f(f(x))=e^x.$$ However I have not still been able to ...
10
votes
2answers
319 views
very elementary proof of Maxwell's theorem
Maxwell's theorem (after James Clerk Maxwell) says that if a function $f(x_1,\ldots,x_n)$ of $n$ real variables is a product $f_1(x_1)\cdots f_n(x_n)$ and is rotation-invariant in the sense that the ...
8
votes
2answers
125 views
Functional Equation: a little tricky
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f[f(x)^2+f(y)]=xf(x)+y$ for all real numbers $x$ and $y$.
Clearly $f(x)=x$ is a solution, check by substitution.
I'm at a loss as ...
6
votes
2answers
248 views
Evaluating $f(x) f(x/2) f(x/4) f(x/8) \cdots$
Let $f : \mathbb R \to \mathbb R$ be a given function with $\lvert f(x) \rvert \le 1$ and $f(0) = 1$. Is there a nice simplified expression for $$\begin{align}F(x) &= f(x) f(x/2) f(x/4) f(x/8) ...
3
votes
1answer
93 views
If $g(x) := \int_1^2 f(xt)dt \equiv 0$ then $f \equiv 0$
Let $f \colon \mathbb R \to \mathbb R$ be a continuous function. Let's
define $$ g(x) := \int_1^2 f(xt)dt. $$
Prove that $g \equiv 0 \Rightarrow f \equiv 0$.
Well, I show you what I have ...
2
votes
1answer
113 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, ...
2
votes
3answers
320 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 ...
1
vote
2answers
849 views
Converting polar equation to cartesian coordinate polar equation and back again?
OK, so I have the following polar equation:
$r = Θ/20$
And I would like to translate this a little to the right, and down from the polar origin.
Now, I figure since I know cartesian coordinate ...
1
vote
2answers
488 views
Solving the functional Equation $f(f(x))=f(x)+x$
Let $f$ be continuous on $\mathbb{R}$. Then how to find all continuous functions satisfying $f(f(x))=f(x)+x$
10
votes
1answer
368 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
$$
\sin^2\theta+\cos^2\theta = 1
...
4
votes
2answers
520 views
How to prove $f(x)=ax$ if $f(x+y)=f(x)+f(y)$ and $f$ is locally integrable
Suppose $f(x)$ is integrable in any bounded interval on $\mathbb R$, and it satisfies the equation $f(x+y)=f(x)+f(y)$ on $\mathbb R$. How to prove $f(x)=ax$?
4
votes
1answer
167 views
$f(x+f(y))=f(x)+y^n$
Here is the problem:
Fix $n\in\mathbb{N}$.
Find all monotonic solutions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
$f(x+f(y))=f(x)+y^n$.
I've tried to show that $f(0)=0$ and derive some ...
3
votes
0answers
143 views
Vector valued contraction
I am familiar with the Banach Fixed Point Theorem and I have used it to prove existence and uniqueness of functional equations in Banach spaces like $C(X)$, the space of bounded continuous function ...
3
votes
2answers
300 views
Implicit function $y = e^{(y-1)/x}$
I'd like to know if the function $ y = f(x) : [0,1] \rightarrow [0,1]$ defined implicitly by the transcendental equation $$\displaystyle y = e^{(y-1)/x}$$ is "well known" (name, properties) or is ...
2
votes
1answer
212 views
About finding the function such that $f(xy)=f(x)f(y)-f(x+y)+1$
Define a function $f\colon\mathbb{R}\to\mathbb{R}$ which satisfies
$$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 ...
