2
votes
0answers
19 views

Show that uniform continuity implies stochastic equicontinuity

Let $\Theta$ be a metric space and assume it is compact. Let $W_t: \Omega \rightarrow \mathbb{R}^k$ be a random variable for $t\leq T$. Let $m(.,\theta): \mathbb{R}^k\rightarrow\mathbb{R}^s$. Let ...
1
vote
2answers
32 views

Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$)

Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$) i) Suppose $\sum a_i$ converges. Must $f'(0)$ exist? ii) Suppose $f(0) = f'(0) = 0$. ...
1
vote
3answers
77 views

Uniform continuity of a function and cauchy sequences

So I'm pretty sure this is almost immediate from the definitions, please tell me if I am incorrect.. Consider two cauchy sequences in D, $\{x_n\}$ and $\{y_m\}$. Since $f$ is uniform continuous we ...
0
votes
0answers
47 views

Continuity of a function that maps non convergent sequences onto non convergent sequences

Let be f a surjective real function defined on R mapping every non convergent sequence onto non convergent sequence. Prove that f is continuous. I have proved that f is injective. Can you help me ...
3
votes
1answer
115 views

If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,…)$, prove that $f(x)=0$ on $[0,1]$.

If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,...)$, prove that $f(x)=0$ on $[0,1]$. This is what I have, how does it look? Proof: Let $P(x)$ be any ...
2
votes
1answer
44 views

$f$ a continuous function, $f^{1/n}$ converges uniformly. How many zeros of $f$?

Let $f$ be a non-negative continuous function on the interval [0,1]. Suppose that the sequence $f^{1/n}$ converges uniformly. How many zeros does $f$ have? I'm confused about what this question ...
1
vote
1answer
62 views

Find a uniformly continuous function such that $a_{n+1}=f(a_n)$

$a_{n+1} = a_n - a_n^2$, $a_1 = 2/3$. for $n\ge1$ a) Show the series converges and find its limit. b) find a uniformly continuous $f:\mathbb{R}\rightarrow \mathbb{R}$ such that: ...
2
votes
1answer
110 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 ...
7
votes
2answers
39 views

Sum of squares at integer points for $L^2$ function

Let $f\in L^2(\mathbb{R})$ be a continuous function such that $f(x)\rightarrow 0$ as $x\rightarrow\pm\infty$. Is it true that $\sum_{n=1}^\infty |f(n)|^2$ is finite? If the continuity and going to ...
3
votes
0answers
29 views

Continuous $L^2$ function has finite sum at integer points?

Let $f\in L^2(\mathbb{R})$ be a continuous function. Is it true that $\sum_{n=1}^\infty |f(n)|^2$ is finite? If the continuity condition is dropped, the statement is not true, because $f(n)$ could ...
1
vote
1answer
113 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 ...
9
votes
1answer
175 views

Is $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ continuous?

Considering the infinite series $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ , I can show that it is not convergent uniformly by Cauchy's criterion and that it is convergent for every $x$ by Dirichlet's ...
3
votes
1answer
194 views

Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum.

Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum. I want to prove this. This is my proof: Since $X$ ...
1
vote
1answer
142 views

Epsilon-delta proof of the existence of the limit of a sequence?

If $\lim_{n\to\infty}a_n \rightarrow L$ and the function $f$ is continuous at $L$, then $$\lim_{n\to\infty}f(a_n) \rightarrow f(L)$$ $\underline{Proof.}$ Let $n, N \in \mathbb{N}$. Let ...
2
votes
0answers
178 views

How to fulfill those boundary conditions?

The problem is the following: Think of a set of functions depending on spherical coordinates given by: $${\psi}_{l m}(r,\theta,\phi) ={k_l}(ar) P_{l}^{m}(\cos \theta) e^{\pm i m\phi} ,$$ so ...
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. ...
2
votes
1answer
68 views

About continuity of functions and limits of sequences

I know that there's a theorem which says that if $f$ is a continuous function, then: $$\lim f(x_n) = f(\lim x_n)$$ This is used to solve, for example: $$\lim(\sin(\frac{2n\pi}{1 + 8n})) = \sin(\lim ...
1
vote
1answer
148 views

Check my work: $\lim a_n = 0 \Rightarrow \lim \sqrt{a_n} = 0 $? (for $a_n$ positive)

I'm trying to prove, as "properly" as possible the following:$$\left[ \lim z_n = z \right] \iff \left[ \lim x_n = x \quad \wedge \quad \lim y_n = y \right]$$ where $z_n = x_n + i y_n$ and $z=x+iy$. ...
2
votes
1answer
73 views

Sequence of continuous functions, integral, series convergence

Let $f_k$ be a sequence of continuous functions on $[0,1]$ such that $\int _0 ^1 f_k(x)x^ndx = \int _0^1 x^{n+k} dx$ for all $n \in \mathbb{N}$. Is $\sum _{k=1} ^{\infty}f_k(x)$ convergent? Could ...
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 ...
4
votes
1answer
165 views

Discontinuous for rationals

Show that $f\left(x\right):=\sum_{n=1}^{\infty}\frac{\left\{nx\right\}}{n^2}$, where $\left\{nx\right\}$ is the fractional part of $nx$, is discontinuous for all rationals. I guess it would be nice ...
0
votes
1answer
56 views

Proof of a special case of Banach's fixed point theorem

I have to prove the following special case of the theorem: Let $f : I \to I$ be Lipschitz continuous on the closed (not bounded) interval $I=[0,\infty)$ with Lipschitz constant $L \lt 1$. Then $f$ ...
1
vote
3answers
304 views

Show that $x\mapsto \frac{1}{x}$ is not uniformly continuous for $x \in (0,1), \frac{1}{x} \in \mathbb{R}$

I am meant to show that for $x \in (0,1)$, $\frac{1}{x}$ is not uniformly continuous. I am able to come up with a proof but the given hint tells me to investigate $x_n = \frac{1}{n}$, $y_n = ...
1
vote
1answer
191 views

Infimum and limit

I was having trouble with the following question. Any help would be highly appreciated. Let $A$ be the set of K-dimensional vectors with non-negative components. Let $B$ be the set of K-dimensional ...
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
183 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 $x \in \left[\frac{1}{2} ,1\right]$ ...
1
vote
2answers
110 views

How to show that $f$ is continuous only at $x=0$

Can any one help me to answer this question: Assuming $$f(x)=\begin{cases} x &\text{if }x\in \mathbb{Q} \\ 0 &\text{if }x\in \mathbb{R}\setminus\mathbb{Q} \end{cases}$$ show that $f$ ...
0
votes
2answers
100 views

Show that $\cos(1 / x)$ cannot be continuously extended to $0$

Can anyone help me to solve this problem? Q: Show that $\cos(\frac{1}{x})$ cannot be continuously extended to $0$? Notice: use this Theorem Let $f:D\rightarrow \mathbb{R}$ and let ...
1
vote
1answer
75 views

question about continuity of a function

everyone can any one solve this problem Q) show that $$f(x)=\begin{cases} x\sin(1/x)&\text{if }x\ne0\\ 0&\text{if }x=0 \end{cases}$$ is continuous on real number? by ...
1
vote
2answers
65 views

question about continuous function

everyone I have question can any one answer prove that $f(x)= |x|$ is continuous on $\Bbb R$? by using the definition of continuous. Thank you
5
votes
1answer
185 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 ...
4
votes
1answer
93 views

Uniform Continuity and partial sums equation proof

Given $f$, a uniformly continuous function defined on the interval $[0,1]$, I need to prove that $$\lim_{n\rightarrow \infty} \frac{1}{2^n} \sum_{k=1}^n (-1)^k \binom{n}{k} f(k/n)=0.$$ I have tried ...
2
votes
1answer
67 views

Continuity of $x+y$ and $xy$ in $\mathbb{R}^{\infty}$

How can I show (or where can I find) that in $\mathbb{R}^\infty$: $f(\textbf{x},\textbf{y})=\textbf{x}+\textbf{y}$, $g(\textbf{x}, k)=k\cdot \textbf{x}$ are continuous functions? ($g$ is from ...
2
votes
2answers
188 views

Continuity of an Analytic Function

I am trying to show that the analytic function, $f: (0, \infty) \to \mathbb{R}$, defined by $ f(x) = \sum \limits_{n = 1}^\infty ne^{-nx} $ is continuous. I don't have much experience with ...
0
votes
1answer
67 views

Maps sending weakly convergent sequences to weakly convergent sequences are continuous?

Well, the question is in the title. I understand that they are continuous in the weak topology, but can't see that it must hold for the norm topology. Please help me.
4
votes
1answer
78 views

A reference for some fact in analysis

I am looking for a reference for the following fact. Any hints would be appreciated. Suppose $(x_n), (y_n)\subset [0,1]$ are some sequences, $(a_n)$ is absolutely summable and for each $f\in C[0,1]$ ...
0
votes
1answer
143 views

Yes or No, Real Analysis, continuity, compactness

Am I correct over statements below? The limsup and liminf of the sequence $n^2$ (meaning $1,4,9,16,\dots$) are equal. T Every bounded sequence has at most one ...
0
votes
1answer
171 views

Let $f$ be a real valued sequentially continuous function relative to a closed bounded interval $I=[a,b]$. Prove that the set $f(I)$ is bounded above

The hint that I've been given is: for each n in the naturals, use the assumption that $n$ is not an upper bound for $f(I)$ to choose a sequence of $x_n$ (from $n=1$ to infinity) in $I$; then apply ...
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 ...
1
vote
2answers
576 views

Show that f is uniformly continuous and that $f_n$ is equicontinuous

$f_n: A \rightarrow \mathbb{R}$,$n \in \mathbb{N}$ is a sequence of functions defined on $A \subseteq \mathbb{R} $. Suppose that $(f_n)$ converges uniformly to $f: A \rightarrow \mathbb{R}$, and ...
3
votes
2answers
250 views

Some homework questions about a Lipschitz function (cauchy sequence)

Do you want to help me with my homework? The exercise is as follows: Consider a Lipschitz function, $h:\mathbb{R}\rightarrow\mathbb{R}$, satisfying for every $x, y$: $$\left| h(x)-h(y) \right| ...
4
votes
1answer
656 views

Continuous Functions and Cauchy Sequences

We know that if a function $f: A \mapsto \mathbb{R}$, $A \subseteq \mathbb{R}$, is uniformly continuous on $A$ then, if $(x_n)$ is a Cauchy sequence in $A$, then $(f(x_n))$ is also a Cauchy sequence. ...