Tagged Questions
3
votes
2answers
57 views
Metric spaces and distance functions.
I need to provide an example of a space of points X and a distance function d, such that the following properties hold:
X has a countable dense subset
X is uncountably infinite and has only one ...
1
vote
1answer
73 views
There exists a continuous function that satisfies this property
Let $X$ be a non-compact subset of $\mathbb{R}$.
I want to show that there a continuous function $f: X \to \mathbb{R}$ such that $f$ is bounded but does not attain its bounds.
I think that there ...
7
votes
2answers
176 views
Give an example of a function $h$ that is discontinuous at every point of $[0,1]$, but with $|h|$ continuous on $[0,1]$
Give an example of a function $h:[0,1]\to\mathbb{R}$ that is discontinuous at every point of $[0,1]$, but such that the function $| h |$ that is continuous on $[0,1]$.
I don't really even know where ...
1
vote
1answer
46 views
If $f_n$ converges uniformly to $f$ on a measure space, show integral of $f_n$ converges to integral of $f$.
Please help me with this problem!
Let $(\Omega,\cal F, \mu)$ be a measure space on which $(f_n)$ is a sequence of bounded, measurable, real-valued functions converging uniformly to $f$.
If ...
3
votes
2answers
139 views
If $f: X \to Y$ is continuous and $E \subseteq X$ is closed and bounded, $f(E)$ is closed and bounded?
I'm trying to think of a counterexample to the claim if $f: X \to Y$ is continuous and $E \subseteq X$ is closed and bounded $f(E)$ is closed and bounded. Here, $X$ and $Y$ are metric spaces. ...
22
votes
2answers
255 views
Is every function with the intermediate value property a derivative?
As it is well known every continuous function has the intermediate value property, but even some discontinuous functions like
$$f(x)=\left\{
\begin{array}{cl}
\sin\left(\frac{1}{x}\right) & x\neq ...
1
vote
0answers
113 views
About $L^p$ membership for different values of $p$
I am struggling in a problem of $L^{p}$ space which is Exercise 6.6 of Folland's Real Analysis (1999), p. 187.
Suppose $0 < p_0 < p_1 \leq \infty$. Find
examples of functions $f$ on ...
1
vote
2answers
75 views
Function oscillating between $[-1,1]$ around $0$.
I was looking for examples of functions $f:\mathbb{R}\to\mathbb{R}$ such that for any $a>0$ we have that $[-1,1]\subseteq f\left([-a,a]\right)$
The two examples I could think of were $\sin(1/x)$ ...
3
votes
4answers
291 views
Examples of functions $f$ and $g$ such that one of them is not differentiable but $f+g$ is differentiable
I'm searching for counterexamples of functions $f$ and $g$ such that one of them is not differentiable but $f+g$ is differentiable. I've already found counterexamples of the product case ($f(x)=|x|$ ...
2
votes
0answers
50 views
Inequality change in $\mathbb{E}[ \max |\cdot|] $ due to $\max$
Let $m$ be a probability measure on $W \subseteq \mathbb{R}^p$, so that $m(W)=1$.
Find $m$, a locally-bounded function $f:\mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R}^p \rightarrow ...
5
votes
1answer
120 views
Is a $C^\infty $ function local monotone at a minimum?
The following is a modified exercise from an analysis 1 book.
Is there a function $f:\mathbb{R} \rightarrow \mathbb{R}$ with:
i) $f$ has in $0$ a strict local and global minimum.
ii) $f\in C^\infty$
...
1
vote
1answer
64 views
finding a function with predetermined $f^{(n)}$s at $0$ and $1$.
Let $a_n,b_n$ be 2 arbitrary sequences of real numbers. Is there any $C^\infty$ function $f:\mathbb{R}\to\mathbb{R}$ such that for any $n$:
$$f^{(n)}(0)=a_n$$
$$f^{(n)}(1)=b_n$$
How about complex ...
6
votes
2answers
183 views
Example of an increasing, integrable function $f:[0,1]\to\mathbb{R}$ which is discontinuous at all rationals?
I have really no idea about this:
Problem: Show that there exists a function $f:[0,1]\rightarrow\mathbb{R}$ such that:
$f$ is discontinuous in all $x\in \mathbb Q$.
$f$ is increasing in ...
3
votes
2answers
125 views
Looking for a bijective nowhere-continuous function ${\mathbb R}\rightarrow{\mathbb R}$
Does there exist a bijective function $f:{\mathbb R}\rightarrow{\mathbb R}$ that is nowhere-continuous, assuming that both domain and range have the "standard topology"? 1
1 By this I mean the ...
0
votes
1answer
22 views
On the existence of functions with a particular convergence
Is the following scenario possible? Provide an example or argue why not.
Let $\{f_n\}_{n=1}^{\infty}$ be measurable non-negative functions on $[0,1]$ converging to $f(x)$ pointwise Lebesgue-almsot ...
3
votes
0answers
31 views
Differentiable function which is nowhere continuously differentiable [duplicate]
Possible Duplicate:
How discontinuous can a derivative be?
$x^2\cos(1/x)$ is the standard example for a differentiable function whose derivative is not continuous at $x=0$.
But is there ...
4
votes
2answers
127 views
Existence of an infinitely differentiable function $ f $ with $ {f^{(n)}}(0) = 0 $ for all $ n \in \mathbb{N} $.
How can one show that there exists an infinitely differentiable function $ f: \mathbb{R} \to \mathbb{R} $ such that $ {f^{(n)}}(0) = 0 $ but $ f^{(n)} \not\equiv 0 $ for all $ n \in \mathbb{N} $?
5
votes
3answers
257 views
If $|f(x)|$ is a differentiable function, then $f(x)$ is also?
If $|f(x)|$ is a differentiable function, then $f(x)$ is also a differentiable function.
Why is this wrong? Can you find a counterexample please?
It seems like a true sentence.
2
votes
1answer
70 views
Any example that $f_n\rightarrow f$pointwise and $f_n'\rightarrow f'$uniformly, but not $f_n\rightarrow f$uniformly?
Let $C$ be an infinite connected set in $\mathbb{R}$ and $\{f_n\}$ be a sequence of differentiable functions from $C$ to $\mathbb{R}^k$.
Suppose
(i)$f_n'$ coverges uniformly $//$
(ii)There exists ...
4
votes
5answers
144 views
irrationality of numbers with rational sum
Assume that $x_1, \dots, x_n$ are non-negative real numbers such that
$$
x_1 + \dots + x_n \in \mathbb Q~~~~~~~~~~~~~~ \text{ and } ~~~~~~~~~~~~~~~x_1 + 2x_2 + \dots + nx_n\in \mathbb Q.
$$
Does ...
1
vote
1answer
29 views
Relations between a product in $L^p$ and essential boundness of a factor
Let be $1\leq p<\infty$ and $g$ a measurable funtion defined on $E$. I have to prove that if $fg\in L^p$ for every $f\in L^p(E)$, then $g$ is essentialy bounded, that is $g\in L^\infty (E)$.
I ...
0
votes
1answer
87 views
Uniform convergence and complete metric space
Let $X$ be a metric space and $\{f_n\}$ be a sequence of functions such that $f_n:E\rightarrow X$.
Suppose $f_n\rightarrow f$ uniformly on a set $E$ and $x$ is a limit point of $E$ and $\lim_{t\to x} ...
3
votes
2answers
342 views
What is an example that a function is differentiable but derivative is not Riemann integrable
I have two questions that i'm curious about.
If $f$ is differentiable real function on its domain, then $f'$ is Riemann integrable.
If $g$ is a real function with intermediate value property, then ...
1
vote
0answers
58 views
Continuous partials at a point without being defined throughout a neighborhood and not differentiable there?
This is a follow-up to Continuous partials at a point but not differentiable there?, but I'll make this question self-contained. Throughout, $f$ will denote a function $\mathbb{R}^2\to\mathbb{R}$. An ...
3
votes
1answer
217 views
Continuous partials at a point but not differentiable there?
In Question on differentiability at a point, it is mentioned (and in Equivalent condition for differentiability on partial derivatives it is cited from Apostol) that for $f:\mathbb{R}^2\to\mathbb{R}$ ...
1
vote
1answer
64 views
Sequence of continuous fuctions $f_n:[0,1]\rightarrow [0,1]$ s.t. $\lim_{n\rightarrow\infty}m(E_n(\varepsilon)) = 0$ but…
Give an example of a sequence of continuous functions $f_n:[0,1]\rightarrow [0,1]$ such that $\lim_{n\rightarrow\infty}m(E_n(\varepsilon)) = 0$ for every $\varepsilon >0$ but ...
0
votes
2answers
56 views
Counterexample of Existence of a continuous extension of a Continuous function
Till now, I have proved followings;
Suppose $X,Y$ are metric spaces and $E$ is dense in $X$ and $f:E\rightarrow Y$ is uniformly continuous. Then,
$Y=\mathbb{R}^k \Rightarrow \exists$ a continuous ...
2
votes
3answers
442 views
Is a bounded and continuous function uniformly continuous?
$f\colon(-1,1)\rightarrow \mathbb{R}$ is bounded and continuous does it mean that $f$ is uniformly continuous?
Well, $f(x)=x\sin(1/x)$ does the job for counterexample? Please help!
1
vote
1answer
229 views
counter-examples in measure theory and set topology
The boundary of a subset of Euclidean space has empty interior, and furthermore has Lebesgue measure zero.Well,this is generally not true,but I can't find an explicit counter-example right now.
...
7
votes
2answers
669 views
convolution of a function with itself equals itself
In a homework question, I was asked to show: (1) in $L^1(R)$, if $f*f = f$, then $f$ must be a zero function. (2) In $L^2(R)$, find a function $f*f=f$. I don't know how to proceed.
for (1), $f*f=f$ ...
3
votes
3answers
306 views
Need some help on a non-example of equicontinuity
In an attempt to better understand the definition of an equicontinuous family of continuous functions, I want to find a simple non-example.
My intuition says that the family ...
2
votes
3answers
136 views
the meaning of “finite” in the finite covering theorem
A textbook I am using to learn analysis states (in reference to just the real line):
Every system of open intervals covering a closed interval contains a finite subsystem that covers the closed ...
0
votes
1answer
104 views
Is this the correct counter example?
I encountered the following problem in Berkeley problems in Mathematics:
(Sp84): Prove or supply a counterexample: If the function $f$ from $\mathbb{R}$ has both a left limit and a right limit at ...
4
votes
0answers
201 views
Fourier dimension of sets of positive Lebesgue measure
Let $K$ be a compact set in $\mathbb{R}$ with positive Lebesgue measure. My question is whether there exists a probability measure $\mu$ supported on $K$ such that $\hat{\mu}(\xi)$, the ...
1
vote
3answers
733 views
second derivative does not exist at specified points
Would any one give me an example or hint how to construct a function whose second derivative does not exist at some specified points say at n number of points. for first derivative I have the modulas ...
3
votes
1answer
72 views
The subset of discontinuous in all points is not open in the space of bounded functions
Let $X\subset \mathcal{B}(\mathbb{R},\mathbb{R})$ be the subset of bounded functions $f:\mathbb{R}\to\mathbb{R}$ such that are discontinuous in all points. Prove that $X$ is not open (with usual ...
17
votes
3answers
322 views
Why does the Hilbert curve fill the whole square?
I have never seen a formal definition of the Hilbert curve, much less a careful analysis of why it fills the whole square. The Wikipedia and Mathworld articles are typically handwavy.
I suppose the ...
5
votes
1answer
168 views
Is my counter-example correct?
In my homework for real-analysis I was asked to prove the following statement:
On $[0,1]$, for $1\leq{}p<\infty$, If $f_{n}\rightarrow{}f$ a.e. and $||f_{n}||_{p}\leq{}M \space\space\forall\space ...
1
vote
2answers
1k views
Multiplying convergent and divergent sequences
When testing for null sequences I've had to say whether they were convergent or divergent, but say you've got a convergent sequence (a) and divergent sequence (b) and you multiplied them (so {ab}) ...
0
votes
1answer
295 views
Real-valued function of one variable which is continuous on [a,b] and semi-differentiable on [a,b)?
Is there any real-valued function of one variable which is continuous on [a,b] and right differentiable on [a,b), but not left differentiable at any point?
4
votes
1answer
384 views
Continuous but not Hölder continuous function on $[0,1]$
Does there exist a continuous function $F$ on $[0,1]$ which is not Hölder continuous of order $\alpha$ at any point $X_{0}$ on $[0,1]$. $0 < \alpha \le 1$.
I am trying to prove that such a ...
39
votes
1answer
1k views
How discontinuous can a derivative be?
There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property. To my knowledge, not many ...
8
votes
2answers
269 views
How do we prove that $\lfloor0.999\cdots\rfloor = \lfloor 1 \rfloor$?
Are the floor functions of $0.999\cdots$ and 1 equal?
It is true that $0.999\cdots=1$ but how does one justifies the integer part of $0.999\cdots$ being 1 , where it is not, or alternatively without ...
2
votes
2answers
203 views
How do you prove the $p$-norm is not a norm in $\mathbb R^n$ when $0<p<1$?
I see that it fails to satisfy the triangle inequality by example but I don't see how to prove this is the case for all $0 < p < 1$. The definition I am using for $p$-norm is $$ \|A\|_p= ...
13
votes
3answers
1k views
Discontinuous linear functional
I'm trying to find a discontinuous linear functional into $\mathbb{R}$ as a prep question for a test. I know that I need an infinite-dimensional Vector Space. Since $\ell_2$ is infinite-dimensional, ...
2
votes
1answer
102 views
Differentiability of a function, of its sections and of its components
Let's consider two cases:
For $f:\mathbb{R}^2 \to \mathbb{R}$, where the domain may be some
open subset in $\mathbb{R}^2$, define its sections to be functions
in $\{ f(,x_2), f(x_1,), \forall ...
-3
votes
1answer
143 views
Examples of interesting sequences [closed]
Any good series or sequences you have?
For example if you sum the reciprocal of primes this diverges.
As Q is de-numerable, almost any Cauchy sequence we pick will not converge in Q.
Stuff like ...
2
votes
1answer
74 views
If $M_0$, $M_1$, and $M_2$ are least upper bounds of $|f(x)|$, $|f'(x)|$ and $|f''(x)|$, does $M_1^2\leq 4M_0M_2$ for vector valued functions?
If $a\in\mathbb{R}$, $f\colon(a,\infty)\to\mathbb{R}$ is a twice-differentiable function, and $M_0$, $M_1$, and $M_2$ are least upper bounds of $|f(x)|$, $|f'(x)|$ and $|f''(x)|$, then ...
12
votes
3answers
384 views
Why does the Continuum Hypothesis make an ideal measure on $\mathbb R$ impossible?
On the page 43 of Real Analysis by H.L. Royden (1st Edition) we read: "(Ideally) we should like $m$ (the measure function) to have the following properties:
$m(E)$ is defined for each subset $E$ of ...
3
votes
1answer
106 views
Dense pre-images implies continuous right inverse?
Suppose $f : \mathbb R \to \mathbb R$ is such that pre-image of every point under $f$ is dense in $\mathbb R$. This, of course, implies that $f$ is surjective, and hence has a right inverse ...
