4
votes
2answers
107 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
46 views

Are there standard parameters for the Weierstrass nowhere differentiable function?

On Wikipedia, the Weierstass non-differentiable function is defined as: $$f(x)=\sum^{\infty}_{n=0}a^n\cos(b^n\pi x)$$ where $0<a<1$, $0<b$, and $ab>1+\frac 32 \pi$ Since it seems like, ...
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
29 views

How to show $\{f_n\}_{n=1}^\infty$ has uniformly convergent subsequence on [0,1]?

Let $\{f_n\}_{n=1}^\infty$ a sequence of second order differentiable functions on the interval [0,1]. If $\forall n\in \Bbb N$ $f_n(0)=f_n'(0)=0$ and for all $n\in \Bbb N$ and $x \in [0,1]$ , ...
2
votes
3answers
65 views

prove that $f(x)=\sum _{n=0}^{\infty}\frac{\cos(nx)}{2^n}$ is continuous

I refered that each fn is continuous because its the fraction of a continuous function by a number and so $f(x)$ that is the sum of continuous functions is continuous. Is it right?
1
vote
2answers
33 views

How would I finish this continuity proof?

I have a multivariable function $f$ with $$f(x, y) = \begin{cases} \frac{x^2+y^2}{y} & \text{if }y \neq 0\\ 0 & \text{if }y = 0 \end{cases}$$ and want to show that it is continuous at $(0, ...
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
20 views

Please verify my work about an equicontinuous sequence

Please check this work below. It is self-explanatory. I am unsure because I use a sequence composed with another sequence with the same index ($f_n^{-1}(u_n)$). We have a sequence of functions ...
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
1answer
29 views

Change of order of limit and function

Let $\Omega\subset\mathbb{R}^n$ be a open and bounded domain. Suppose that $f(x)$ is a $C^1$ function for $x\in\Omega$ and $\{ x_k \}_{k=1}^{k=\infty}\in\Omega$ is a sequence with ...
2
votes
0answers
23 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
38 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
97 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 ...
3
votes
1answer
152 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
45 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
68 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
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 ...
7
votes
2answers
40 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
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 ...
9
votes
1answer
242 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
205 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
144 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
181 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
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. ...
2
votes
1answer
69 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
151 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
92 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
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 ...
5
votes
1answer
175 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
60 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
342 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
228 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
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
223 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
120 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
107 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
76 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
68 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
208 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
64 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
97 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
68 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
205 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
74 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
151 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
189 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 ...