1
vote
1answer
43 views

For continuous functions, preimage of open set is open.

Let $f$ be a continuous function from a metric space $X$ into $Y$. If $V\subset Y$ and $V$ is open, then show that $f^{-1}(V)$ is open. The proofs I've seen of the fact that open sets have open ...
0
votes
2answers
39 views

Continuous function vs Uniformly continuous function

Can you give me an example of the function in metric space which is continuous but not uniformly continuous. Definitions are almost the same for both terms. This is what I found on wiki: ''The ...
0
votes
1answer
38 views

No direct proofs of “if $ f: (X, d_X) \to (Y, d_Y)$ is continuous and $X$ is compact then $f$ is uniformly continuous.”

I am studying the theorem "if $f:(X,d_X)\to (Y,d_Y)$ is continuous and $X$ is compact, then $f$ is uniformly continuous." I am not looking for a proof, but I have an argument against any attempt at a ...
3
votes
2answers
51 views

Why the continuity of a function on a metric space doesn't depend on metrics?

In the definition of the continuous function on a metric space, it seems to me that a continuous function depends on the metric of the given metric space. Could somebody explain Why the continuity of ...
4
votes
3answers
42 views

Does continuity of $f$ imply $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$?

I'm struggling to prove or disprove that the continuity of $f$ implies $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$. $f:X\to Y$ is a map between metric spaces $(X,d),(Y,d')$ while $\bar M$ denotes the ...
3
votes
0answers
37 views

Sufficient conditions for closed infinite pasting lemma

It's well known that the pasting lemma for infinitely many closed sets is false. It's reasonably easy to cook up examples such that for $X = \bigcup X_i$ with $X_i$ closed in $X$ such that $\left. ...
2
votes
1answer
60 views

Proving the metric attains a minimum on a compact subset

Let $(X,d)$ be a complete metric space. Suppose $B \subset X$ is compact. Prove that for every $a\in X$ the minimum $\min_{b\in B} d(a,b)$ exists. I'm pretty sure you can do this by just using the ...
1
vote
0answers
46 views

Characterization of continuity by subsequences.

I have a difficulty trying to prove the following proposition. Any help would be greatly appreciated. $\textbf{Prop.}$ Let $(X,d_1)$ and $(Y,d_2)$ be two metric spaces. A function $f:X\rightarrow Y$ ...
1
vote
1answer
43 views

Let $f$ be a continuous functions from $[0,1]$ to $\mathbb{R}$. Then, $f$ is not necessarrily lipschitz.

Let $f$ be a continuous functions from $[0,1]$ to $\mathbb{R}$. Then, $f$ is not necessarily lipschitz. Is the above statement true? I thought since $f$ is continuous on a compact metric space, $f$ ...
0
votes
3answers
58 views

Compact Space: Locally Continuous $\implies$ Uniformly Continuous

Given metric spaces. Prove that any locally continuous function on a compact space is uniformly continuous!
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votes
1answer
57 views

Show for $f:A \to Y$ uniformly continuous exists a unique extension to $\overline{A}$, which is uniformly continuous

Working on the following problem from Munkres: Let $(X, d_{X})$ and $(Y, d_Y)$ be metric spaces; let $Y$ be complete. Let $A \subset X$. Show that if $f:A \to Y$ is uniformly continuous, then ...
0
votes
1answer
98 views

A continuous function on the real line such that the preimage of every point is either empty of has exactly 3 points [duplicate]

Let $f : \mathbb{R} \to \mathbb{R}$ be a function with all fibres $(\lbrace{x \in \mathbb{R}| f(x) = c\rbrace} = f^{−1}(c)$, $c \in \mathbb{R})$ either empty or consisting of exactly three points. ...
3
votes
1answer
48 views

What does it means that sequences characterize closed sets and functions?

A text book I'm reading says at one point the following: "In metric spaces are sequences the ones which chacterize closed sets and continuous functions". What is exactly the meaning of that ...
0
votes
2answers
29 views

Something not working out for me in the continuity definition

I'm studying analysis and I've ran into this proposition saying that a function from a metric space X to a metric space Y, is continuous if and only if for every open set O in Y, the inverse image of ...
0
votes
0answers
43 views

Onto continuous function on a compact metric space is isometry. [duplicate]

Let $K$ be a compact metric space with metric $d$ and suppose $f:K\rightarrow K$ is continuous and surjective (onto), and satisfies $d(f(x),f(y))\leq d(x,y),\,\forall x,y\in K$. How can we prove that ...
1
vote
1answer
35 views

continuous functions on metric space

Assume $f:X\rightarrow Y$, where $X$ and $Y$ are two metric spaces. If $f(\overline{E})\subset \overline{f(E)}, \, \forall E\subset X$, then how can we prove that $f$ is continuous? Thank you for ...
0
votes
2answers
33 views

Lipschitz does not imply fixed points

I have the following problem in mind: Let us say we have a function $f:X\rightarrow X$ (X is a complete metric space) and it respects that if $x\neq y$ then : $d(f(x),f(y))<d(x,y)$. My trouble ...
0
votes
1answer
32 views

Prove $g$ is continous on metric space.

For:$(X,d)$ is a metric space , $f:X\to X$ is a continous function and $g:X\to R, x\mapsto g(x)=d(f(x),x)$. Prove that $g$ is a continous function. Definition: $f$ is continous at $x_0$ if only if ...
4
votes
2answers
92 views

Is $\mathbb R^{\omega}$ homeomorphic to $\mathbb R^{\omega} \times \mathbb R^{\omega}$?

As a study exercise, I'm trying to find a topological space $X$ which is homeomorphic to $X \times X$. I began thinking of simple examples involving $\mathbb R$ but then realized my best bet would be ...
3
votes
3answers
78 views

Supposed X and Y are metric spaces and $f:X \rightarrow Y$ and $g:X \rightarrow Y$ are continuous prove $\{x \in X:f(x)=g(x)\} $ is a closed set

Supposed X and Y are metric spaces and $f:X \rightarrow Y$ and $g:X \rightarrow Y$ are continuous prove $\{x \in X:f(x)=g(x)\} $ is a closed set I don't have any idea where to start. Any suggestions? ...
3
votes
2answers
100 views

Why is pointwise continuity not useful in a general topological space?

On page 27 of Lee's Introduction to Topological Manifolds, he writes In metric spaces, one usually first defines what it means to be continuous at a point...in topological spaces, continuity at a ...
3
votes
0answers
49 views

Sufficient conditions for existence of injection from a metric space $M$ to $\mathbb{R}$

Let $M$ be any metric space. What conditions are required of $M$ for there to exist an injective, continuous function $$\varphi \colon M \longrightarrow \mathbb{R}$$ I would like to believe that ...
1
vote
2answers
52 views

Continuity from complete metric space

Let $f:X\rightarrow Y$ be a continuous function, such as: $f(X)=Y$. If $X$ is complete, does it imply $Y$ is complete?
0
votes
1answer
27 views

Continuity under different metrics

On the real line R define the standard Euclidean metric d(x,y)=|x-y| and e(x,y)=|arctan(x)-arctan(y)|. Show that: (i) A function f:R→R is continuous under d if and only if it is continuous under e;
1
vote
1answer
74 views

addition and multiplication of functions in function space, continuous?

I have a norm that works in function space of C[0,1]. How do I show that addition and multiplication of functions (C[0,1]xC[0,1]->C[0,1]) are continuous functions?
1
vote
3answers
138 views

Can $\le$ be used insted of < in the definition of continuity?

A common definition of a continuous map $T:M_1\to M_2$ is that for every $x\in M_1$ and every $\epsilon>0$ there exists a $\delta >0$ such that for all $y$ in $M_1$ $$d_1(x,y)<\delta \implies ...
0
votes
1answer
73 views

Topology, continuous mapping help!!

Give an example of a continuous mapping $$f:X \to Y; \ \ (X,\ Y\ \text{metric})$$ for which there exists an open subset $U \subset X$ such that $$f(U)=\{\ y \in Y \ \mid \ \exists x \in U: ...
5
votes
1answer
110 views

Is every $G_\delta$ set the set of continuity points of some function $f$?

I can prove that given a function $f:X \rightarrow Y$, where $X,Y$ are metric spaces, the set $A \subseteq X$ of points on which $f$ is continuous, is $G_{\delta}$. (Take $U_n = \bigcup_{y \in ...
1
vote
2answers
43 views

Analogue of closed graph theorem

This is the analogue of closed graph theorem for compact space Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph of $f: X \rightarrow K$ is a closed subset ...
0
votes
1answer
35 views

Detail on a theorem of continuity and compactness

I have come across a proof in a book. I have trouble convincing myself on a statement on said proof. The theorem is a well-known one. I am stating the version I found on Ross's "Elementary Analysis." ...
1
vote
1answer
50 views

About the continuity of a function in the closed graph theorem proof

I'm reading Functional Analysis book of Rudin, and in the proof of the closed graph theorem, there's one point that I don't understand. Can someone please explain it to me? I really appreciate this. ...
2
votes
2answers
59 views

if a map and its inverse are continuous, does that imply injection?

I've proved that a mapping of one topology to another and its inverse are both continuous. so since f and f inverse are continuous, can I therefore say that they're injective?
1
vote
4answers
64 views

Prove B is a closed subset of X given the f and g are continous?

Let $(X;\rho)$, $(Y;\sigma)$ be metric spaces. Let $f,g : X \to Y$ be continuous. Prove that the set $B=\{x\in X: f(x)=g(x)\}$ is a closed subset of $X$
2
votes
2answers
146 views

Prove that the identity map $(C[0,1],d_1) \rightarrow (C[0,1],d_\infty)$ is not continuous

$$d_\infty = \max|x_i - y_i|$$ $$d_1 = \sum_{i=1}^n |x_i - y_i|$$ The first part of this question was to prove that the identity map $$(C[0,1],d_\infty) \rightarrow (C[0,1],d_1)$$ is continuous, ...
0
votes
1answer
36 views

$\sup_n \inf_{y \in X} (F(y) + n d(x,y))= F(x)$

Let $(X,d)$ be a metric space and $F : X \rightarrow [0, +\infty)$ a lower semicontinuous function. Then $$ \sup_n \inf_{y \in X} (F(y) + n d(x,y))= F(x). $$ Is this true? Intuitively it works since ...
1
vote
3answers
82 views

Continuity of metric [duplicate]

I recently came across this definition: Let $(X,d)$ be a metric space and $A$ be a nonempty subset of $X$. For each $x\in X$ we define a distance from $x$ to $A$ by the equation $d(x,A)=\inf\{d(x,a) ...
1
vote
1answer
153 views

Show that $f:[0,1] \to [0,1]$ is continuous if $f(x) = x^{1/k}$ for any $k \in \mathbb N$

I'm very confused right now and I want to apply the theorem that says " A mapping f of a metric space $X$ into a metric space $Y$ is continuous on $X$ if and only if $f^{-1}(V)$ is open in $X$ for ...
1
vote
1answer
39 views

Calculus continuity question.

show that the function f(x,y)= |x-1| + |y-1| is continuous at (2,2) using epsilon delta definition. The way I have done this is as follows. |f(x,y)-f(2,2) = ||x-1|+|y-1|-(1+1)| = ||x-1|+|y-1|-2| ...
2
votes
1answer
146 views

Proving that a holder continuous function always has a smaller exponent.

According to wikipedia if we have $f:X \rightarrow Y$ which is $\alpha$-Holderian then for all $\beta < \alpha$ the function is also $\beta$-Holderian. How do we prove this starting from the fact ...
1
vote
1answer
26 views

Checking of continuity

While compactness & connectedness are preserved under continuous maps, this question comes to my mind: $f : \mathbb R \to \mathbb R$ is strictly monotone increasing function such that {$ f(x) : x ...
2
votes
1answer
49 views

Proving that half an isometry is a homeomorphism

Let $(K,d)$ be a compact metric space and $f:K\rightarrow K$ such that $$\forall x \in K, \forall y \in K, d(f(x),f(y)) \geq d(x,y)$$ Prove that $f$ is a homeomorphism. What I managed to prove is ...
3
votes
1answer
38 views

Isn't the result true for any $A\subset X?$

There's a problem in my text which reads as: Let $f: (X, d)\to(Y, d)$ be continuous. Let $A\subset X$ be open. Show that the restriction $f|_A$ of $f$ to $A$ is a continuous function from the metric ...
3
votes
1answer
118 views

$O(n,\mathbb R)$ of all orthogonal matrices is a closed subset of $M(n,\mathbb R).$

Let $M(n,\mathbb R)$ be endowed with the norm $(a_{ij})_{n\times n}\mapsto\sqrt{\sum_{i,j}|a_{ij}|^2}.$ Then the set $O(n,\mathbb R)$ of all orthogonal matrices is a closed subset of $M(n,\mathbb ...
1
vote
1answer
65 views

Show that for $n\in\mathbb N$ the mapping $f:M(n,\mathbb R)\to M(n,\mathbb R):A\mapsto A^n$ is continuous.

Show that for $n\in\mathbb N$ the mapping $f:M(n,\mathbb R)\to M(n,\mathbb R):A\mapsto A^n$ is continuous. ($M(n,\mathbb R)$ is identified with $\mathbb R^{n^2}$ as a normed liner space.) ...
0
votes
2answers
140 views

continuous mapping is determined by its values on a dense subset of its domain

Question: If f and g are continuous mappings of a metric space X into a metric space Y, let E be a dense subset of X. if g(p) = f(p) for all p $\in$ E, prove that g(p)= f(p) for all p$\in$ X. Answer: ...
3
votes
1answer
189 views

Zeroes of a continuous function on a metric space

Let $f$ be a continuous real valued function on a metric space $X$. Let $Z(f)$ be the set of all $p\in X$ such that $f(p)=0$ $\text{(a)}$ Prove that $Z(f)$ is closed. $\text{(b)}$ Recall ...
1
vote
2answers
37 views

Continuity and Metric Spaces

How do I show that the function $f:X \to \mathbb R$ given by $$f(x)=\frac{d(a,x)}{d(a,b)}$$ is continuous. Given that $(X,d)$ is a metric space, and $a,b$ are distinct points in $X$.
0
votes
0answers
59 views

Distance function on metric space is continuous [duplicate]

In showing that the diameter of a compact set $A$ is attainable, one approach is to consider a function $f:A\times A\rightarrow\mathbb{R}$ such that $f(x,y)=d(x,y)$. The key is to show that the ...
2
votes
1answer
101 views

Show that the function $f: X \to \Bbb R$ given by $f(x) = d(x, A)$ is a continuous function. [duplicate]

I'm studying for my Topology exam and I am trying to brush up on my metric spaces. Suppose $(X, d)$ is a metric space and $A$ is a proper subset of $X$. Show that the function $f: X \to \Bbb R$ given ...
3
votes
1answer
425 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 ...