All Questions
Tagged with metric-spaces continuity
991
questions
0
votes
0
answers
69
views
The function $d \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ defined by $d\big((a,b)\big)=\lvert x-y\rvert$ is continuous [duplicate]
Let the function $d \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ be defined by
$$
d\big( (x, y) \big) := \lvert x-y \rvert \qquad \mbox{ for all } (x, y) \in \mathbb{R}^2.
$$
Let $\mathbb{R}$ and $...
- 26.9k
-2
votes
1
answer
69
views
Rudin Ch 4 exercise 3: the zero set of a continuous function is closed
Let $f$ be a continuous real function on a metric space $X$. Let $Z(f)$ (the zero set of $f$) be the set of all $p \in X$ at which $f(p) = 0$. Prove that $Z(f)$ is closed.
My attempt
Let $p$ be a ...
- 1
1
vote
1
answer
34
views
If $f:X \rightarrow Y$ is an $\epsilon$-map between compact metric spaces then there exists a $\delta > 0$ such that $diameter(f^{-1}(Z)) < \epsilon$
First a map $f: X \rightarrow Y$ is called an $\epsilon$-map if it is continuous, onto and for any $y \in Y$, $diameter(f^{-1}(y)) < \epsilon$. Now I'm trying to find a $\delta > 0$ such that if ...
- 802
0
votes
0
answers
35
views
Proving $g(x) = 1 − \frac{\lVert x−p\rVert}{\delta}$ is continuous
I want to show that given any two points $p, q \in K$ with $p\neq q$ we can choose
a continuous function $g \in C(K)$ so that $g(p)\neq g(q)$, specifically by letting $g(x) = 1 − \frac{\lVert x−p\...
- 477
0
votes
0
answers
22
views
Two injective paths $\gamma_1$ and $\gamma_2$ with the same curve can be expressed as $\gamma_1 = \gamma_2 \circ \alpha$ for some $\alpha$
My professor provided a proof to the following theorem that I am not quite able to understand. Any help would be greatly appreciated since I already spent a lot of time deciphering this (not very long)...
0
votes
0
answers
38
views
Assuming Euclidean Metric in $\mathbb{R}$
I came across an exercise asking to prove that for a metric space $(X,d)$, with $p \in X$, the distance to p is a continuous map. I.e. that $f: X \rightarrow \mathbb{R}$ defined by $f(x) = d(x,p)$ for ...
- 35
0
votes
2
answers
36
views
Prove or disprove $C:=\{x\in\mathbb{R}^n|f(x)\leq f(x_0)\}$ is closed
Let $f:\mathbb{R}^n\to\mathbb{R}$. Given $x_0\in\mathbb{R}^n$, define
$$C:=\{x\in\mathbb{R}^n|f(x)\leq f(x_0)\}.$$
Show that $C$ is closed.
My attempt:
Let $\{x_n\}$ be a sequence in $C$ that ...
- 249
0
votes
1
answer
45
views
Subset of $C^1([0 , 1])$ is open
Let $C^1([0 , 1],\mathbb{R})$ be the subspace of $C([0 , 1],\mathbb{R})$ of the functions that have a continuous derivative throughout $[0 , 1]$ with norm:
$\|f\|=\sup_{x \in [0,1]} \{ |f(x)|+|f'(x)| \...
0
votes
0
answers
46
views
The derivative of a function of several variables
Define a function $f:\mathbb{R}^2\to\mathbb{R}$ by: $$f(x,y)=\begin{cases}
\dfrac{x\sin y - y\sin x}{x^2+y^2}\ &\text{if}\ (x,y)\neq(0,0)\\0\ &\text{if}\ (x,y)=(0,0)
\end{cases}.$$
I want to ...
- 159
0
votes
0
answers
49
views
Left and right continuity in metric spaces only for $f:\mathbb{R}\longrightarrow\mathbb{R}$?
Let $(M,d_{M})$ and $(N,d_{N})$ be metric spaces and let $f:M\longrightarrow N$ be a function;
Metric space definition of continuity: $f$ is continuous at a point $x\in M$ if for every $\epsilon>0$ ...
- 1,010
2
votes
1
answer
92
views
What properties of metric spaces are not preserved by uniformly continuous isomorphism?
Compactness and connectedness are preserved by homeomorphism, in the sense that if two metric spaces $(X,d_X)$ and $(Y,d_Y)$ are homeomorphic and $(X,d_X)$ is compact then it follows that $(Y,d_Y)$ is ...
- 23
1
vote
2
answers
64
views
A doubt on a problem involving continuous function on a compact metric space
Let $X$ be a compact metric space with metric $d$ and let $f \in C (X, X)$
and such that $d (f(a ),f(b))\ge d (a, b)$ for all $a$ and $b$ in $X.$ Show that
$d(f(a), f(b)) = d(a, b)$ for all $a$ and $...
- 779
0
votes
1
answer
71
views
Example to show that the containment $\overline {f^{-1}(B)} \subset f^{-1}(\bar B) $ is proper where $f$ is continuous mapping
Let $f: X \to Y$ be a continuous function, where $B\subset Y$.
Then,
$\overline {f^{-1}(B)} \subset f^{-1}(\bar B) $ holds, here's the proof.
I am looking for an example to illustrate that the above ...
- 1,239
1
vote
0
answers
55
views
Continuity of quotient map with respect to topology induced by metric
Let $C([0,T];\mathbb{R}^d)$ denote the space of continuous functions with the usual supremum norm and given a path $x\in C([0,T];\mathbb{R}^d)$ let $x^s$ denote the stopped path $x(t \land s)$ with $t\...
- 934
1
vote
1
answer
80
views
compact metric space and the existence of f(A)=A
Let $X$ be a compact metric space , $f: X\to X$ be a continuous map.
Prove that there exists a non-empty subset $A\subset X$ s.t $f(A)=A$.
My attempt:
Set $A_1=f(X), A_n=f(A_{n-1}), n\geq 2$. Then $\{...
user1277306