5
votes
Accepted
Uniqueness of Ring Homomorphisms from $\mathbb{R}$ to $M_2(\mathbb{R})$
Whoops, I misspoke in the comments; even $f(1) = I$ is not enough to guarantee uniqueness if you don't require that $f$ is $\mathbb{R}$-linear. The reason is that $M_2(\mathbb{R})$ contains $\mathbb{C}...
- 441k
5
votes
Accepted
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}, f(f(x))+f(f(y))+2f(xy)=f(x^2+y^2)$.
Here is a self-contained, full solution.
Step 1. Let $\mathsf{FE}_1(x, y)$ denote the original functional equation
$$ \color{blue}{\mathsf{FE}_1(x,y)} : \quad f(f(x)) + f(f(y)) + 2f(xy) = f(x^2 + y^2)...
- 172k
3
votes
Accepted
Solve the functional equation for all functions defined under non-negative integers
You have $(m-n)f(m+n)=0$. So if $k$ can be written as a sum of two distinct parts, then $f(k)=0$. The only natural which cannot be written this way is zero. This means $f(n)=0$ for $n>0$. Setting $...
- 8,246
2
votes
Replacing $x$ in Functional Equation
With the given functional equation $2f(x)+f(1-x)=x^2$ we can choose any value of $x$ and we should expect the equation to be true (since we're told that it holds for any real value of $x$).
If we set $...
- 26.1k
2
votes
Uniqueness of Ring Homomorphisms from $\mathbb{R}$ to $M_2(\mathbb{R})$
You have edited your question to assume that $f(1)=I_2$. The question is actually easy if $f(1)$ is singular instead.
The three properties imply that $f(1)$ is nonzero and idempotent. If $f(1)$ is ...
- 142k
1
vote
How to solve $f(x)=f(0)+\frac{x}{f'(x_{0})+\frac{x^{2}}{f''(x_{0})+\ddots}}$ for $f(x)$?
(Several remarks, too long for a comment)
I have tried to solve your problem for a while now, but unsuccessfully unforturnately. Nonetheless, here are a few remarks, which might help you to restate ...
- 11.7k
1
vote
Accepted
Discontinuous solutions of a functional equation involving n-simplices
Definitions: Let
$$
\mathcal G = \{(v_0,v_1,\ldots,v_n) \in (\mathbb R^n)^{n+1}:\{v_1-v_0,\ldots,v_n-v_0\} \text{ is a basis of } \mathbb R^n\}
$$
be the set of $(n+1)$-tuples of vectors in $\mathbb R^...
- 6,134
1
vote
Putnam 1988 A5 Question
For $a_0=x$ we define the sequence $a_n$ by $a_{n+1}=f(a_n).$ Then $a_{n+2}=6a_n-a_{n+1}.$ Assume $a_n$ is positive. We are going to show that $a_1=2x,$ i.e. $f(x)=2x.$
Let $h_n={a_{2n+1} \over a_{2n}...
- 36.1k
1
vote
Find all continuous functions $f : \mathbb{R} \to \mathbb{R}$ satisfying $f(z) \leq 1$ for some $z \neq 0$
The only constant solutions are
$$\boxed{\text{S1 : }f(x)=1\quad\forall x},$$
which indeed fits, and
$$\boxed{\text{S2 : }f(x)=-\frac{1}{2}\quad\forall x},$$
which also fits. So, let us from now on ...
- 31
1
vote
Find all continuous functions $f : \mathbb{R} \to \mathbb{R}$ satisfying $f(z) \leq 1$ for some $z \neq 0$
(This is not a complete answer.) First of all, in your case $1$, when setting $y = 0$, you forgot a minus sign and your equation for $f$ is actually for all $x$, $2f(x)^2 + \frac{1}{4} + f(x)^2 = 1$ ...
- 7,161
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