Tagged Questions
6
votes
1answer
72 views
Existence of a power series converging non-uniformly to a continuous function
I am wondering whether there exist a function $f(z) = \sum_{n\geq0} a_n z^n$ such that:
$f$ converges and is continuous on the closed unit disk $D$ and
the series $\sum_n a_n z^n$ does not converge ...
1
vote
1answer
20 views
Equicontinuity and uniform convergence 2
Let $\{f_n\}_n$ be a sequence of real valued functions on a compact metric space $K$. Suppose that for all $x$ we have $f_n(x) \to f(x)$ as $n \to \infty$ and that the family $\{f_n\}_n$ is ...
3
votes
2answers
65 views
Do $L^2$ convergence and continuity imply pointwise convergence?
It is said here that $L^2$ convergence and continuity imply pointwise convergence (just before paragraph $5.2$) but I can't find how to prove it. Does anyone see how ?
1
vote
2answers
38 views
Convergence of $\max_{0\le i\le n}|f(i/n)|$
Suppose that $f\colon [0,1]\to\mathbb R$ is a continuous function. How can I prove that
$$\max_{0\le i\le n}\biggl|f\Bigl(\frac in\Bigr)\biggr|\to\sup_{0\le x\le1}|f(x)|$$
as $n\to\infty$?
Any help ...
3
votes
5answers
121 views
Sequence of continuous functions which converges to a continuous limit
Any help with this: construct a sequence of continuous functions defined on $ [0,1] $ which converges pointwise but not uniformly to a continuous limit ?
Thank you.
0
votes
2answers
50 views
Composition of a continuous function with functions that converge uniformly
Here is problem that appeared in one of the past final exams for my introductory real analysis course, that I am having hard time to solve. It is Question 5 in 8 of the following file:
...
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 $(f_n):[−1,1]\rightarrow \mathbb{R}$ ...
5
votes
2answers
106 views
Need to prove $f$ continuous at $x_0$ iff for every monotonic sequence $(x_n)$ converging to $x_0$ we have $\lim f(x_n)=f(x_0)$
This was a problem that the Professor went over in class, but I am having trouble understanding and finishing the proof. The full question is:
$f:I \rightarrow \mathbb R$ is continuous at $x_0 \in I$ ...
0
votes
1answer
22 views
Understanding this theorem about continuity at $c$ and a sequence converging to $c$
I want someone to explain to me just this part:
Let $f:D\rightarrow \mathbb{R}$ and let $c\in D$. Then $f$ is continuous at $c$ if and only if, whenever $X_n$ is a sequence in $D$ that converges ...
1
vote
3answers
60 views
Math Analysis - Problem with showing sequence of functions is convergent and uniformly convergent
Let $f:\left[\frac{1}{2} ,1\right] \rightarrow \mathbb R$ be a continuous function, $\{g_n\}_{n=1}^{\infty}$ a sequence of functions where $g_n(x) = x^n f(x)$, with $x \in \left[\frac{1}{2} ,1\right]$ ...
3
votes
1answer
31 views
Baire one functions, closed intervals
I've been wondering if you could help me with the following problem.
There's an article on Baire one functions (number 2 on google search list) and there is one thing concerning Lemma 9 that I am not ...
2
votes
1answer
29 views
Baire one functions, characteristic functions of intervals
Do you think you could help me prove that characteristic functions of intervals are Baire one functions?
And is it true that linear combinations of Baire one functions are also Baire one?
5
votes
1answer
59 views
Pointwise limits of continuous functions
Could you help me prove the following?
Let S be the set of function that are the pointwise limit of continuous functions, $\{h _n\} \subset S$ with max$_{x \in [0,1]} |h_n(x)|< A_n$ and $\sum ...
5
votes
0answers
61 views
Functional sequence [duplicate]
Let $(f_n)$ be a sequence of functions $\mathbb{R} \rightarrow \mathbb{R}$.
Suppose that for any $(x_n)$ convergent to $x$ we have $f_n(x_n) \rightarrow f(x)$.
Prove that $f$ in continuous, there is ...
1
vote
1answer
128 views
Uniform convergence of continuous functions with Lipschitz limit
Let $K \subset \mathbb R^d$ be a compact. Let $\phi_{\varepsilon} \colon K \rightarrow \mathbb R$ be continuous and converge uniformly to $\phi$. Suppose further that $\phi$ is Lipschitz continuous. ...
1
vote
1answer
65 views
Uniform contraction proof
Prove that for every uniform contraction function $f$ there exists a
unique real $z$ such that $f(z)=z$. A function $f:\mathbb R\to\mathbb R$ is called a uniform contraction if there exists an $a$ in ...
4
votes
1answer
117 views
convergence of nets vs. convergence of really long sequences
It is well known that for a function $f:X\to Y$ between the underlying sets of topological spaces, the condition that $f$ is continuous is equivalent to the condition that given any net $N$ in $X$ ...
3
votes
2answers
365 views
Dini's Theorem. Uniform convergence and Bolzano Weierstrass.
In Spivak's chapter on uniform convergence he asks to prove the following
THEOREM Let $\{f_n\}$ be sequence of continuous functions that converge pointwise to $0$ over $[a,b]$. If $0\leq ...
3
votes
1answer
323 views
{$\int_{[1/n,1]}f$} to converge and yet $f$ is not $L$-integrable over $[0,1]$
Let $f$ be a function on $[0,1]$ and continuous on $(0,1]$.
I want to find a function $f$ s.t. {$\int_{[1/n,1]}f$} converges and yet $f$ is not $L$-integrable over $[0,1]$.
My attempts:
I've found ...
1
vote
2answers
127 views
Continuity in Frechet spaces
These are undoubtably simple questions, but I have no background in functional analysis and am wondering about them. The first is an exercise from Folland, the second is not, but both are claims I've ...
1
vote
1answer
55 views
a.s discontinuous on an interval but continuous in Probability?
Can a Gaussian process be almost surely discontinuous on an interval T but at the same time be continuous in Probability everywhere on T?
Alternative question: can a sequence of discontinuity points ...
4
votes
1answer
88 views
uncountable subset of $C[0,1]$ has uniformly convergent subsequence
If $S$ is an uncountable subset of $C[0,1]$, then there is a uniformly convergent sequence $\{f_n\}$ of distinct functions of $S$.
I know how to do this for $C^1[0,1]$ since $S \subset
\cup_{m,n ...
0
votes
1answer
155 views
Convergence in measure and pointwise convergence in continuity points
Hi can you help me with the following:
$\{f_n\}$ a sequence of increasing functions with $f_n\to f$ in measure on $[a,b]$.
Show that $f_n(x)\to f(x)$ at every $x$ where $f(x)$ is continuous.
...
14
votes
2answers
480 views
If $\lim_n f_n(x_n)=f(x)$ for every $x_n \to x$ then $f_n \to f$ uniformly on $[0,1]$?
This is a self-posed question, so I do not know the answer and I would like to know what do you think about.
Let $f,f_n:[0,1]\to \mathbb R$ be continuous functions. Assume that for every sequence ...
1
vote
1answer
151 views
Alternative conditions for Uniform Convergence
Let $\{f_v\}_v\in\mathbb N$ be a sequence of continuous functions $f_v:\Re^m\to\Re^n$ and $f:\Re^m\to\Re^n$ and assume that $f_v$ converges to $f$ pointwise (i.e. For every fixed $x\in\Re^m$, ...
4
votes
2answers
185 views
The relationship between Fourier coefficients of function $f$ and its continuity
How to prove that if Fourier series of function $f$ converge uniformly, then function is continuous?
