Tagged Questions

3
votes
1answer
43 views

Proving Continuity with Open Sets

I have a doubt about how to prove continuity using the definition in terms of open sets. The $\epsilon$-$\delta$ definition of continuity is not very pleasant to work with, however, I know what must ...
1
vote
2answers
42 views

Finding a bounded, non-compact set of functions $f:[0,1]\to\Bbb R $

Consider the metric space $(X, d)$ given by $$X = \{\text{all continuous functions}\,f:[0,1]\to\Bbb R\}$$ with $$d(f,g)=\sup_{t\in[0,1]}|f(t)-g(t)|.$$ Find with proof a set $A \subseteq X$ with ...
0
votes
2answers
49 views

Find a convergent function in metric space

Let $C[−1, 1]$ be the space of continuous functions equipped with the metric $p(f,g) = \max\{|f(x)−g(x)| \mid x \in [−1, 1]\}$. Then the sequence of functions $(f_n):[−1,1]\rightarrow \mathbb{R}$ ...
4
votes
2answers
38 views

Intuitive explanation of ball-based definition for continuity of functions in metric spaces

First of all, hat tip to @Fayz for providing this definition. Backstory: I broke my glasses several days ago and, in the meantime, this important definition was written on a board I could not see. ...
0
votes
2answers
43 views

Need to confirm: Sup Metric $C[0,1]$, question about boundary

For the sup metric, $C[0,1]$. Let $S \subset C[0,1]$ be given by: $$S=\left\{f:[0,1]\to \mathbb{R} \ : \ 0 \leq f\left(\frac{1}{2}\right)<1\right\}$$ The question is simple: is this set open or ...
2
votes
1answer
74 views

Continuity in metric space, TRUE or FALSE?

Let $(X,d)$ and $(Y,e)$ be metric spaces , and let $f: X \to Y$ be a function. True or false ? Give a proof or a counterexample as appropriate. $(a)$ If $d$ is the discrete metric on ...
3
votes
3answers
56 views

Continuous map between metric spaces

Suppose $X,Y$ are metric spaces, let $A \subset X$ be a bounded subset of $X$ and $f: A \to Y$ to be a continuous bjection. Prove or disprove that $f^{-1}$ is continuous. Remark: If each closed ...
0
votes
1answer
39 views

Let $ f:(X, d) \mapsto (Y,d) $ be an mapping such that $ Graph (f) $ is connected. [duplicate]

Where $ X $ is connected. Does it imply $ f $ to be continuous?
3
votes
1answer
54 views

How to show that a continuous map on a compact metric space must fix some non-empty set.

Suppose $(X,d)$ is a compact metric space and $f:X\to X$ a continuous map. Show that $f (A)=A$ for some nonempty $A\subseteq X.$ I start this by supposing that $A_0:=X$ and $A_{n+1}:=f(A_n)$ for ...
5
votes
4answers
80 views

Is a continuous function like a homomorphism/isomorphism for metric spaces?

If I had to define a notion of a homomorphism/isomorphism on metric spaces, I'd say something like this. Let $A$ and $B$ be metric spaces with norms $\| \cdot \|_A$ and $\| \cdot \|_B$ respectively. ...
4
votes
1answer
66 views

Why $f(x) = \frac{d(x,A)}{d(x,A)+d(x,B)}$ is uniform continuous?

Let $X$ be a metric space, $A$ and $B$ are two subsets of $X$. $d(x, A) = \inf_{z \in A}d(x,z)$ and $\inf_{x \in A,y \in B}d(x,y) = \delta > 0$ We define $$f(x) = \frac{d(x,A)}{d(x,A)+d(x,B)}$$ ...
1
vote
1answer
24 views

Compact metric space: proof $\text{diam}(K)$

I am to assume that $K$ is a compact metric space. I must prove that there are two points $x,y$ contained in $K$ such that $d(x,y)=\text{diam}(K)$. Recall $\text{diam}(K)= \sup \{ d(x,y) \mid x,y ...
0
votes
1answer
41 views

$f:X\rightarrow X$ be a continuous map, we need to show $f(\cap A_n)=\cap f(A_n)$

let $X$ be a complete metric space with metric $d$ and $A_{i}$'s are nested sequence of closed sets in $X$ i.e $[A_1\supseteq A_2\dots]$ such that $\sup\{d(x,y):x,y\in A_n\}\to0$ as $n\to\infty$ ...
2
votes
1answer
41 views

equivalent metric

Let $(X; d)$ and $(Y; d')$ be metric spaces, and let $f : X \to Y$ be continuous. Define $df (x; y) = d(x; y) + d'(f(x); f(y))$ for $x, y \in X$. Show that $df$ is a metric on $X$ that is equivalent ...
4
votes
3answers
70 views

Consequence of Invariance of Domain

The Invariance of Domain theorem states that Given a continuous injection $f : U \to \mathbb{R}^n$, where $U$ is a nonempty open subset of $\mathbb{R}^n$, $f$ is an open map. These slides (see ...
4
votes
2answers
93 views

Continuous linear functionals

Let L be a continuous linear functional on a metric linear space X. Prove: L(S) is a bounded set for any bounded subset S of X. The metric is translation invariant.
1
vote
2answers
88 views

Bounded functions on subsets of Euclidean space

It is known that given any closed and bounded $X \subseteq \mathbb{R}^n$ and a bounded continuous function $f : X \to \mathbb{R}$, $f(X)$ has a minimum value and maximum value. This can be proved by ...
1
vote
1answer
92 views

Extension of continuous function

The question is: Let $(K,\rho)$ be compact metric space. $F\subset K$ closed. $f:F\rightarrow \mathbb{R}$ continuous. Is there a continuous extension of $f$ on $K$? Attempt: Suppose there exists ...
2
votes
1answer
69 views

Continuity of metric space of integrals of continuous functions

Let $R$ be the real line with the standard metric $d:R \times R \to R$ be defined by $d(x,y) = |x-y|$. Let $X$ be the set of continuous functions $f:[a,b] \to R$ of an arbitrary closed interval ...
6
votes
1answer
179 views

pointwise limit on a complete metric space

Let $\{f_n: X\rightarrow \mathbb{R}\}$ be a sequence of continuous real-valued functions on a complete metric space, $X$. Suppose this sequence has a pointwise limit, $f$. How easy is it to see that ...
1
vote
1answer
99 views

Characterising continuous maps between metric spaces

Let $f:(X,d)\to (Y,\rho)$. Prove that $f$ is continuous if and only if $f$ is continuous restricted to all compact subsets of $(X,d)$. I could do the left to right implication but couldn't do the ...
2
votes
2answers
98 views

Metric Of A Graph

The following is question 6 from page 99 of Walter Rudin's Principles Of Mathematical Analysis. I'm having trouble understanding what the metric of the graph might be (which, as far as I can tell, is ...
0
votes
2answers
97 views

Finding a continuous function with specified properties

This is a homework question in my analysis class: Let $A$ and $B$ be two nonempty closed subsets of a metric space $X$ that do no intersect. Show that there is a continuous function $f:X\rightarrow ...