7
votes
Accepted
Function that satisfies $f(x-f(x))=f(x)$
This solution is due to a colleague o'mine.
Let $g_n(x):\equiv x - n f(x)$. As you have noticed, we have $f(g_n(x))\equiv f(x)$, $\forall n\in\mathbb N$.
Lemma. The function $g_n:\mathbb R\to\mathbb ...
- 5,643
4
votes
Why is a function not differentiable at the end-points of its domain?
Is there a source telling you that you can't say $x^2$ on $[0,8]$ is differentiable at the endpoints?
The way you pose the question makes me suspect you are studying the Mean Value Theorem, which is ...
- 50.8k
2
votes
Accepted
Continuity of point to set mapping
Let $N$ be a neigbhorhood of $\bar x$ such that $\text{cl}(\Omega(N))$ is compact, where $\Omega(N) := \bigcup_{x\in N} \Omega(x)$. Because the sequence $x_k \to \bar x$, it is eventually inside $N$, ...
- 45k
2
votes
Help with proving that a function is increasing and continuous
What I write here might not be the simplest way to handle this, but I would think of it like this.
First, if $\phi$ is monotone increasing, we can prove that $\phi(x^-) = \sup \{\phi(y) : y < x\}$ ...
- 1,925
2
votes
Accepted
Help with proving that a function is increasing and continuous
Let $x\in [0,1)$ and let $\epsilon>0$.
By the continuity of $f$, there exists $\delta>0$ such that $|f(x)-f(y)|<\epsilon$, whenever $y\in (x-\delta,x+\delta)$.
We are going to show that $|\...
- 3,548
1
vote
Help with proving that a function is increasing and continuous
I'd use the definition. Let $\varepsilon >0$. You need to show that exists $\delta>0$ such that $\forall x\in(a-\delta,a+\delta), x\neq a$ we have $\varphi(x)\in(\varphi(a)-\varepsilon,\varphi(a)...
- 11
1
vote
Help with proving that a function is increasing and continuous
A way, f because it is continuous in [0,1] has a maximum in $x_0$ then $\varphi (x)=f(x_0), \ if \ x \ \in [x_0,1]$, then $ \varphi $ is continuous in $(x_0,1]$ for being constant.$\varphi$ is right ...
- 58
1
vote
Accepted
Why is a locally Hölder continuous and bounded function pointwise Hölder continuous?
Let $f$ be $\alpha-$Holder continuous on $D$. I will assume that $D$ is a domain.
Let $x_{0}\in D$. There exists $\delta>0$ such that
$B_{\delta}(x_{0})$, the ball centered at $x_{0}$ with radius $\...
- 3,548
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