1
vote
1answer
19 views

Continuity of a map from the 2-plane.

Let $f: \mathbb{R}^{2} \rightarrow X$ be a map where $X$ is a Hausdorff topological space. Assume that the restriction of $f$ on $\mathbb{R}^{2}-\{0\}$ is continuous, and the restriction of $f$ on any ...
4
votes
2answers
106 views

$\lim_{x\rightarrow 1}\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^x}=\ln2$.

Prove $$\lim_{x\rightarrow 1}\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{{n}^{x}}=\ln2.$$ Of course $$\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{{n}}=\ln2,$$ but we can not use the Proposition : If a ...
0
votes
1answer
25 views

In the semi linear uniform space

In the semi linear uniform space, If $f$ is a function from $(X ,Γ_X)$ to ($Y,Γ_Y)$ where $f(x_n)$ converges to $f(x)$ whenever $x_n$ converges to $x$,show that $f$ is continuous at $x$.
1
vote
1answer
22 views

Sequence problem dealing with continuity and convergence.

I need help in this question. I figured out a way to solve the question but not sure the proof is valid. This is the question, Given $a \in\mathbb{R}$, and a function ...
1
vote
2answers
57 views

sequential continuity vs. continuity

A short and hopefully simple question for someone with more experience in topology: If a topology is induced by a mode of convergence and in fact nothing more is known apriori, wether if this topology ...
0
votes
0answers
30 views

A function is discontinuous at all rational points and continuous at all irrational points [duplicate]

Define $f(x)$ for $x\in[0,1]$ by $f(\frac pq)=\frac1q$ if $p$ and $q$ are relatively prime, and $f(x)=0$ if $x$ is irrational. How can we see that $f$ is discontinuous at all rational points and ...
0
votes
1answer
42 views

What am I doing wrong in this continuity check?

I want to show that the function $f$ is discontiunous. $f$ is defined as follows: $$f(x,y) := \begin{cases} (x^2+y^2)\cdot\sin(\frac{1}{\sqrt{x^2+y^2}}) & , (x, y) \neq (0,0) \\ 0 ...
0
votes
2answers
67 views

Prove $f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ defines a continuous function on $\mathbb{R}$.

Prove $$f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k$$ defines a continuous function on $\mathbb{R}$. I think we can show that if $\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ is uniformly ...
2
votes
0answers
25 views

A basic question about upper hemicontinuity

Given a correspondence $f:X\rightarrow 2^X$, suppose X is a closed simplex in $\mathbb{R}^n$, and $f$ is compact-valued. We say $f$ is upper hemicontinuous if, $\forall x\in X$ and every open subset ...
0
votes
1answer
40 views

Are these $f_n$ equicontinuous?

Let $f_n$ be a sequence of real-valued functions defined on $\mathbb{R}$ satisfying $f_n \to f$ uniformly in the compact subsets of $\mathbb{R}$ $f_n^{-1}$ is bi-Lipschitz $1 \leq (f_n^{-1})'(x) ...
2
votes
2answers
90 views

Properties of the function defined by $g(x) = \sum\limits_{n=0}^{\infty} \frac{1}{1+n^2x^2}$

I am looking at the function $g:\mathbb{R} \rightarrow \mathbb{R}$ defined as $$g(x) = \sum\limits_{n=0}^{\infty} \frac{1}{1+n^2x^2}$$ I would like to know if this function is convergent, continuous ...
1
vote
0answers
30 views

Prove that the limit of the function a sequence is the same as the function of the limit of the sequence [duplicate]

Assume that $\lim_{n\to \infty}a_n=a.$ Suppose that the function $f$ is continuous everywhere including at $a$. Form the sequence $(f(a_n))_{n=1}^{\infty}$. Prove that $\lim_{n\to ...
0
votes
0answers
22 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
73 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
66 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
45 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 ...
6
votes
1answer
343 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
47 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
55 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
128 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
89 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
341 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
40 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
40 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
44 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
125 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
55 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
64 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
57 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
119 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
71 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
194 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
191 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
70 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
93 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
117 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
176 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 ...
4
votes
1answer
123 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
124 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
56 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
238 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.
2
votes
2answers
266 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
67 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
399 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
34 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
220 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
41 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
64 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
207 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 ...