All Questions
Tagged with metric-spaces calculus
242
questions
0
votes
3
answers
58
views
Prove $f$ does not attain its minimum
Let $C[0,1]$ be a family of continuous functions on $[0,1]$ with
$||x||_{\infty}=\sup\limits_{t\in [0,1]}|x(t)|$. Define $$A=\{x\in
C[0,1]: x(1)=1, -1\leq x(t)\leq 1\}.$$
$a)$ Prove $f:C[0,1]\to\...
- 249
0
votes
2
answers
34
views
Prove $f$ attains its maximizer [duplicate]
Let $f:\mathbb{R}^n\to\mathbb{R}$ be a continuous function. 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.
Suppose that $f$ is coercive in ...
- 249
0
votes
1
answer
22
views
Analysis of Metric Properties in an Infinite Set with Discrete Metric
I am not sure if my solution to the following problem is correct
Let x be an infinite set. for x $\in$ X and y $\in$ X we define:
\begin{equation}
d(x,y) = \left\{
\begin{array}{ll}
1 & x \...
- 21
0
votes
0
answers
85
views
NBHM 2023 Question : Mark true /false
Let $f$ be a real-valued function on the interval $[0, 1]$ such that:
$
f(\lambda x + (1 - \lambda) y) \leq \lambda f(x) + (1 - \lambda) f(y) \quad \text{for all } x, y, \lambda \in [0, 1].
$
(a) The ...
- 376
2
votes
0
answers
81
views
If every continuous function $f:K \rightarrow \mathbb{R}$ achieves its maximum and minimum on $K$, is $K$ compact? [duplicate]
A well known result from mathematical analysis is that if $K$ is a non empty compact metric space and $f:K \rightarrow \mathbb{R}$ is a continuous function, then $f$ achieves its maximum and minimum ...
- 751
1
vote
2
answers
161
views
Continuity iff lower/upper semi continuity (using $\epsilon$-$\delta$)
The definitions that I am using are:
A real valued function $f$ defined on a metric space X is said to be continuous at $x$ iff for each $\epsilon > 0$ there exists $\delta > 0$ such that if $|x-...
- 796
0
votes
1
answer
27
views
The space of equall $L_1$ vectors
I am trying to understand norms. I have seen this common picture where we visualize the norms.
If I read it well. These curves, express the vectors with $L_p$ norm equal to $1$.
So if I wanted, for ...
- 829
1
vote
1
answer
38
views
Showing the unboundedness of the function $f(x)$
$
f(x) = \begin{cases}
\frac{xy^2}{x^2+y^6} & \text{falls } (x, y)^t \neq (0, 0)^t \\
,0 & \text{sonst}
\end{cases}$
Show that ${g|_{B_r(0)}}$ is unbounded for every${(r > 0)}$ and show ...
- 11
0
votes
1
answer
47
views
Proving or disproving continuity of integral operator on $(C^0([0, 1])$ in $ |\cdot|_\infty$ Norm
$\text{Consider } (C^0([0, 1]), |\cdot|_\infty). \text{ Then, } T: C^0([0, 1]) \rightarrow \mathbb{R} \text{ with } T(f) = \int_0^1 f(x)^2 , dx \text{ is continuous.}$
I guess that it's not correct ...
- 1,324
0
votes
1
answer
46
views
Proving the continuity of a logarithmic function in a metric space
Let $(X, d)$ be a metric space, $a \in X$, and consider the function $f: X \setminus \{a\} \to \mathbb{R}$ given by
$f(x)=log(d(x,a)).$
Then $f$ is continuous. Prove or disprove it.
My idea:
Let $(X, ...
- 1,324
0
votes
0
answers
62
views
Let $(X, d)$ be a metric space, and let$f: X \to \mathbb{R}$ be a continuous function.If$A \subseteq X$is bounded, then necessarily $f(A)$ is bounded. [duplicate]
Prove or disprove:
Is my solution correct? I assume that the statement is correct, but I'm not sure how to prove it:
Let $(X, d)$ be a metric space, and let $f: X \to \mathbb{R}$ be a continuous ...
- 1,324
1
vote
1
answer
178
views
Proof of a metric space axiom for $d(f,g)=\int_{a}^{b}|f(t)-g(t)|dt$
In a topology textbook, I found the following exercise:
Let $X$ be the set of all continuous functions $f:[a,b] \to \mathbb{R}$.
For $f,g \in X$, define $$d(f,g)=\int_{a}^{b}|f(t)-g(t)|dt.$$
Using ...
1
vote
2
answers
35
views
Does $x_1$ belong to $\overline{B}(y, \gamma_2)$?
Let $(X, d)$ be a metric space and let $0<\gamma_1<\gamma_2$. Let $x_1, x_2\in X$ and $y\in X$ be fixed. Assume that
$$ x_2\in \overline{B}(y, \gamma_2)\quad\text{ and }\quad d(x_1, x_2)\le \...
user603537
0
votes
1
answer
29
views
If $I$ is a bounded interval then f is lipschitzian. [duplicate]
I'm having trouble finding the lipschitzian constant from the exercise below for $n$ even or odd, the constant will depend on $n$ and also on the limitation of $f$. Can someone help me please ?
Let $f ...
- 13
0
votes
2
answers
103
views
If $A\subseteq\mathbb{R}^n$ is compact and $x\in\mathbb{R}^m$ then $A\times \{x\}$ is compact.
I'm stuck in this part of the Heine Borel proof. Could you help me please? I'm trying to prove it using the fact that a compact set is such that given a open cover we can extract a finite subcover. ...
- 1