Tagged Questions
3
votes
1answer
58 views
Cauchy's functional equation - a generalisation? (do additive maps have to be continuous?)
If a map $f : \mathbb{R} \to \mathbb{R}$ is additive, in the sense that $f(x + y) = f(x) + f(y)$, then it is simple to show that $f$ is $\mathbb{Q}$-linear, buy it does not need to be ...
9
votes
2answers
66 views
Uniqueness of solution for a functional equation
Let $h \in \mathcal{C}:=C([-a,a])$, where $a>0$. Prove that there exists a unique function $f \in \mathcal{C}$ such that
$$
f(x)=\frac{x}{2}f\Big(\frac{x}{2}\Big)+h(x)\quad \forall x \in [-a,a].
$$
...
8
votes
1answer
138 views
A functional equation problem
Let $f$ be a function which maps $\mathbb{Q}^{+}\to\mathbb{Q}^{+}$. And it satisfies
$$
\left\{ \begin{array}{l}
f(x)+f\left(\frac{1}{x}\right)=1\\
f(2x)=2f(f(x))
\end{array}\right.
$$
Show that ...
2
votes
1answer
86 views
How bad can perturbing a functional equation really make things?
A long time ago, I found occassion to find solutions to a functional equation of the following form
$f(x-y) = f(x) - f(y) + \delta $ with $\delta \in \mathbb{R}.$
Using the same exact techniques as ...
5
votes
3answers
158 views
Find the value of the function at the given point.
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function satisfying the conditions
$$\begin{align*}
(1)&f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}\\
(2)&f(0)=1\\
(3)&f'(0)=-1
...
17
votes
2answers
357 views
Proving that $f(n)=n$ if $f(n+1)>f(f(n))$
How can we prove that if $f:\mathbb{N}\rightarrow\mathbb{N}$ is a function so that $f(n+1)>f(f(n))$ for all $n\in\mathbb{N}$ then $f(n)=n$ for all $n\in\mathbb{N}$?
1
vote
1answer
94 views
A functional equation related to the exponential function
Suppose that $f:\mathbb{R}\rightarrow (0,\infty)$ and $g:\mathbb{R}\rightarrow
\mathbb{R}$ are two functions and satisfies the following relation
\begin{equation}
f(xy)=f(y)^{g(x)}
\end{equation}
...
2
votes
0answers
98 views
Convexity conditions for $f$ and $\dfrac {1} {f}$
Let $f:\mathbb{R} _{+}\rightarrow \mathbb{R} _{+}$ be a real function.
Find all conditions on $f$ under which
$f$ is convex on $\left( a;b\right) \subset \mathbb{R} _{+}$$\Leftrightarrow$ $\dfrac ...
1
vote
3answers
147 views
Solution of functional equation $f(x)=-f(x-a)$
I have a problem with finding solution. I suppose it will be something like $f(x) =G(x)\Re(e^{\frac{x\pi}{a}})$, where $\Re$ is real part of a complex number, $G(x)$ periodic function whith period ...
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 $c \in \mathbb{R}$ ...
3
votes
4answers
324 views
If $f(x + y) = f(x) + f(y)$ showing that $f(cx) = cf(x)$ holds for rational $c$
For $f:\mathbb{R}^n \to \mathbb{R}^m$, if $f(x + y) = f(x) + f(y)$ for then for rational $c$, how would you show that $f(cx) = cf(x)$ holds?
I tried that for $c = \frac{a}{b}$, $a,b \in \mathbb{Z}$ ...
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
...
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, ...
4
votes
2answers
515 views
$f(x) + f(1-x) = f(1)$
I was discussing with a friend of mine about her research and I came across this problem.
The problem essentially boils down to this.
$f(x)$ is a function defined in $[0,1]$ such that $f(x) + f(1-x) ...
0
votes
2answers
127 views
How to solve the following system?
I need to find the function c(k), knowing that
$$\sum_{k=0}^{\infty} \frac{c(k)}{k!}=1$$
$$\sum_{k=0}^{\infty} \frac{c(2k)}{(2k)!}=0$$
$$\sum_{k=0}^{\infty} \frac{c(2k+1)}{(2k+1)!}=1$$
...
1
vote
0answers
168 views
How to find a function which satisfies such functional equation?
How to find a function which satisfies:
$$a^x=\lim_{h\to\infty} \left( f_a \left(f_a^{[-1]}(x)+\frac xh\right)\right)^{[h]}$$
where $f^{[n]}(x)$ is the number if iterations of a function (if n=-1 ...
1
vote
3answers
115 views
Solving Series of equations
I have the following series of equations (n+2 equations n+2 variables):
$k_0q_0+\lambda q_0 + c_0 = 0$
$k_1q_1+\lambda q_1 + c_1 = 0$
$k_nq_n+\lambda q_n + c_n = 0$
$q_1+q_2+....+q_n = 1$
...
0
votes
1answer
162 views
Solving this set of quadratic equations
I have a set of quadratic equations of the form..
$ 2S_0q_0(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_0 = 0$
$ 2S_1q_1(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_1 = 0$
.
.
.
$ 2S_nq_n(q_0S_0 + q_1S_1 ...
0
votes
1answer
148 views
Finding $\alpha$ such that $f(\alpha(x+y))=f(x)+f(y)$
Problem taken from the link: http://web.mit.edu/rwbarton/Public/func-eq.pdf I am stating the question here
For which $\alpha$ does there exists a nonconstant function $f: \mathbb{R} \to \mathbb{R}$ ...
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 $f(f(x))=f(x)+x$
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 ...
8
votes
1answer
359 views
Polynomials which satisfy $p^{2}(x)-1 = p(x^{2}+1)$
Can we find a polynomial $p(x) \in \mathbb{R}$ such that $\text{deg}\ p(x)>1$ and which satisfies $$p^{2}(x)-1=p(x^{2}+1)$$ for all $x \in \mathbb{R}$.
This question can be very well identified with ...
