0
votes
0answers
17 views

Sufficient conditions for Uniform Law of Large Numbers

I would need a Uniform Law of Large numbers for $f_T(\theta)$ over $\Theta$ when $f$ is the indicator function and, thus, not continuous over $\Theta$. Do you know about any sufficient conditions?
2
votes
0answers
68 views

Convergence of a sequence pointwise a.e. on a manifold given that it converges on a reference manifold

Let $\Gamma_t$ be a compact hypersurface for each $t \in [0,T]$. Let $Q=\Gamma_0\times(0,T)$ and $$Q_T :=\bigcup_{t \in (0,T)} \Gamma_t \times \{t\}.$$ For each $t \in [0,T]$, suppose that ...
1
vote
1answer
47 views

Showing if $f_n \to f$ uniformly and each $f_n$ has at most $10$ discontinuities, then so does $f$

Suppose that $f_n:[a,b] \to \Bbb R$ and $f_n$ uniformly converges to $f$ as $n$ goes to infinity. How to prove that if each $f_n$ has at most ten discontinuities (the discontinuities for each $f_n$ ...
1
vote
1answer
32 views

Lower semi-continuity of a convex functional on $L^1(\Omega,[0,1])$

Let $\Omega$ be a bounded domain and $f:\Omega\times[0,1]\to[0,\infty]$ be such that $x\mapsto f(x,u)$ is measurable for every $u$, $u\mapsto f(x,u)$ is continuous and convex for a.e. $x$. Furthermore ...
3
votes
1answer
137 views

Equicontinuity implies uniform convergence

So I know it's a theorem that if $\{f_n\}$ is a sequence in an equicontinuous family of functions defined on a compact metric space $K$ then if for all $x$, $f_n(x)\rightarrow f(x)$ pointwise then ...
2
votes
2answers
45 views

how to prove that the limit of this sequence of functions is continuous?

I have a norm that works in function space and that is $‖∙‖_{sup}:C([0,1])→R$, $‖∙‖_{sup}:=sup${$|f(t)|$}. I need to show that the metric is complete. So I need to show that every Cauchy sequence of ...
3
votes
2answers
51 views

minimum/maximum of two limits

Let $f: \mathbb {R} \rightarrow \mathbb {R} $ continuous so that $v =\lim\limits_{x \rightarrow -\infty} f(x) $ and $w = \lim\limits_{x \rightarrow \infty} f(x)$ are existing. I want to show that ...
2
votes
1answer
107 views

$x_n$ convergence to $x$ implies $f_n(x_n)$ convergence to $f(x)$. prove that $f$ is continuous

Let $f$ and $f_n$ be functions from $\mathbb{R} \rightarrow \mathbb{R}$ Assume that $f_n (x_n) \rightarrow f (x)$ as $n\rightarrow \infty$ whenever $x_n \rightarrow x$. Prove that $f$ is ...
0
votes
2answers
63 views

If $(f_n)$ converges pointwise to $f$ on $\mathbb{R}$, does this imply that $(f_n)$ uniformly converges to $f$ on an interval of $\mathbb{R}$?

If a sequence $(f_n)$ converges pointwise to the same function $f$ on all $\mathbb{R}$, does this imply that $(f_n)$ uniformly converges to $f$ on an interval of $\mathbb{R}$? From what I understand, ...
0
votes
4answers
146 views

Sequence of continuous functions whose pointwise limit is discontinuous

Give an example of a sequence of continuous functions $f_n:\mathbb{R} \to \mathbb{R}$ such that the pointwise limit $f:\mathbb{R} \to \mathbb{R}$ exists but is discontinuous. My idea is ...
0
votes
2answers
38 views

If each $f_n$ is continuous on a set $S$, does $f_n$ converge pointwise to a function $f$ on $S$?

If each $f_n$ is continuous on a set $S$, does $f_n$ converge pointwise to a function $f$ on $S$?I feel I am seriously misunderstanding something. Am I asking a vacuous question?
3
votes
1answer
35 views

$L^2$ norm convergence with continuity assumption [duplicate]

Let $f,f_1,f_2,\ldots\colon\mathbb{R}\rightarrow\mathbb{R}$ be continuous functions in $L^2(\mathbb{R})$. Suppose that $\|f_n-f\|_2\rightarrow 0$ as $n\rightarrow 0$. Is it true that ...
2
votes
1answer
43 views

Trying hard to see the fault in my proof - though it is incorrect!

consider the function $ f(x) = \dfrac{1}{1-x} $ on the interval $[0,1)$ My claim is that the function is bounded in this interval (even though it's not), I'm asking if someone could tell me where ...
1
vote
1answer
80 views

For a continuous function $f$ and a convergent sequence $x_n$, lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$

Let $f:X \rightarrow Y$ be a function. Prove that if $f$ is continuous, then for every convergent sequence $(x_n)$ lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$ My ...
1
vote
1answer
49 views

Pointwise Convergence and Continuity

I am having trouble knowing how to start a homework problem. If anybody could give me the first step, or lead me through a different but similar example, that would be greatly appreciated. The problem ...
1
vote
1answer
62 views

convergence of compact sets

Let $X$ and $Y$ be compact sets (both subsets of the real line). Assume $Y$ has a non-empty interior. Consider the continuous function $f:X \rightarrow Y$. For any given $y$ in the interior of Y, and ...
2
votes
1answer
47 views

Convergence of compact sets continuous function

Let $X$ and $Y$ be compact set (both subset of the real number). Consider the continuous function $f:X \rightarrow Y$. For any given $y$, and for $h>0$ small enough so that $y+h \in Y$. I want to ...
1
vote
1answer
112 views

Show $\sum^\infty_{n=1}(\frac{x}{n^{0.6}(1+nx^2)})$ converges uniformly on $\mathbb{R}$

$\sum^\infty_{n=1}\frac{x}{n^{0.6}(1+nx^2)}$ converges uniformly on $\mathbb{R}$ Is $x\rightarrow\sum^\infty_{n=1}(\frac{x}{n^{0.6}(1+nx^2)})$ continuous at all points of $\mathbb{R}$? I'm stuck on ...
3
votes
3answers
68 views

$f(x)=\sum_{n=0}^{+ \infty} \frac{(-1)^n}{(n!)^2}\left( \frac{x}{2}\right)^n $ is continuous

Let \begin{align} f: \begin{cases} \mathbb{R} &\longrightarrow \mathbb{R} \\ x & \longmapsto \displaystyle \sum_{n=0}^{+ \infty} \frac{(-1)^n}{(n!)^2}\left( \frac{x}{2}\right)^n\end{cases} ...
3
votes
1answer
151 views

Show convergence of a sequence of continuous functions $f_n$ to a continuous function $f$ does not imply convergence of corresponding integrals.

Let $f_n\in C([0,1])$ be a sequence of functions converging uniformly to a function $f$. Show that $$\lim_{n\rightarrow\infty}\int_0^1f_n(x)dx = \int_0^1 f(x)dx.$$ Give a counterexample to show that ...
1
vote
2answers
43 views

Let $\sum_{n=1}^{\infty}n^5(\frac{x}{x+2})^n=S(x)$. Prove that the sum S(x) is a function and continuous to $x\epsilon [0,10]$

Let $\sum_{n=1}^{\infty}n^5(\frac{x}{x+2})^n=S(x)$. Prove that the sum S(x) is a function and continuous to $x\epsilon [0,10]$ Since we are talking about sums and we need to prove continuous i ...
0
votes
3answers
165 views

Show that $f$ is discontinuous.

Let the sequence of function $f_{n}=\sqrt[2n+1]{x}$ (for $x\geq 0$). I've shown that it converges pointwise to $f$, that is $$\lim_{n\to\infty}f_{n}(x)=f(x)=\left\{\begin{matrix} 0 ...
0
votes
1answer
69 views

Check the continuity of the next function $f(x)=\sum_{n=1}^{\infty}(x+\frac{1}{n^2})^n$

Check the continuity of the next function $f(x)=\sum_{n=1}^{\infty}(x+\frac{1}{n^2})^n$ I've started by doing Cauchy test to check when the sum converges: ...
2
votes
1answer
88 views

Proving that $f(x)=\sum^{\infty}_{n=1}\frac{1}{n^2-x^2}$ is continuous.

Prove that $f(x)=\sum^{\infty}_{n=1}\frac{1}{n^2-x^2}$ is continuous at all $x \notin \Bbb N$. An attempt: We should consider showing that $\sum^{\infty}_{n=1}\frac{1}{n^2-x^2}$ converges uniformly. ...
3
votes
2answers
103 views

Berkeley exam summer '79, sequence of continuous functions, integral, convergence

I've recently been browsing some Berkeley exams and I'm particularly interested in Problem 19 here. Let ${f_n}$ be a sequence of continuous real functions defined on $[0,1]$ such that $\int_0^1 ...
6
votes
1answer
163 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 ...
3
votes
1answer
96 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
113 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
49 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
207 views

Sequence of continuous functions which converges to a continuous limit [duplicate]

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.
1
vote
2answers
218 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
65 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
368 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
33 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
182 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
40 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 ...
3
votes
1answer
56 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
184 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
62 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
278 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
91 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
190 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 $X$ ...
5
votes
2answers
1k 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
4k 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
264 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
80 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 ...
5
votes
1answer
128 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
182 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
725 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
220 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 $x\in\Re^m$, ...