1
vote
1answer
34 views

Are the family of given nice functions $f\subset C^0(I,[0,1])$ equicontinuous?

The family of continuous functions $f\in\mathcal{F}$ are defined on a closed subset of real numbers $I\subset\mathbb{R}$ as follows: \begin{equation} f(y) = \begin{cases} 0, &l(y)<\rho \\ ...
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 ...
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?
0
votes
0answers
14 views

Uniform law of large numbers for discontinuous functions?

Do you know about any Uniform Law of Large numbers (see http://en.wikipedia.org/wiki/Law_of_large_numbers#Uniform_law_of_large_numbers) that work when f is the indicator function (and thus not ...
0
votes
0answers
28 views

If periodic function has a discontinuity at $x_0$ its Fourier series cannot converge uniformly on any interval containing $x_0$, why?

I'm reading about Fourier series and in one point my book states the following: Suppose $f$ is a periodic function. If $f$ has a discontinuity at $x_0$, the Fourier series of $f$ cannot ...
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 $f_n$ ...
0
votes
0answers
54 views

Composition of uniform convergent function is not uniform convergent

I am trying to come up with an example for the following situation: Say we have 2 sequences of functions $f_n:U \rightarrow R $ and $g_n: \rightarrow W $ both uniformly convergent to $f$ resp. $g$ ...
0
votes
1answer
96 views

Uniformly continuous function approximated by Lipschitz

Let $f~:~\mathbb R \to \mathbb R$ a uniformly continuous function. How could I construct a sequence $(f_n)_{n\ge 0}$ of Lipschitz functions such that $(f_n)$ uniformly converges towards $f$?
4
votes
0answers
47 views

Why is this Takagi's function continuous?

1903 Takagi constructed the function $f: [0,1] \rightarrow \mathbb{R}$ with $f(x) := \sum_{k=0}^\infty 2^{-k} \mathrm{dist}(2^k x, \mathbb{Z})$ where $\mathrm{dist}(x,A) := \inf\{|x-y| : y \in A\}$ ...
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, ...
1
vote
2answers
154 views

Pointwise convergence does not imply $f_n(x_n)$ converges to $f(x)$

I have been given a sequence of real valued continuous functions $(f_1,f_2,...,f_n)$, and a real valued sequences $(x_1,x_2,...,x_n)$, where $x_n$ converges to $x$. Also $f$ is a continuous real ...
3
votes
0answers
70 views

Equicontinuity and Uniform Boundedness

If we have a sequence of smooth functions $\{f_{n}\}_{n}$ where $f_{n}: U \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^{n}$. We are given the following two results: For $x \in U$ we have ...
0
votes
1answer
80 views

uniform convergence

I'm trying to understand the notion of uniform convergence, say that I have a squence of functions $(f_n(x)= (1 -\frac{x}{n}\sin(x))^{-n})_{n \in \mathbb{N}}$. We know that it converges to ...
2
votes
2answers
72 views

Uniform Convergence; Continuity

In the last step. I believe we did $$\frac{d}{dx} \int_{a}^{x} g = \frac{d}{dx} \lim_{n \to \infty} \int_{a}^{x} f_n' = \lim_{n \to \infty}\frac{d}{dx} \int_{a}^{x} f_n' = \lim_{n \to \infty} ...
0
votes
1answer
173 views

Proving uniform convergence of $g(f_n(x))$

Let $f(x),f_n(x):[a,b]\to[c,d]$ s.t $f_n\to f$ uniformly on [a,b]. Let $g:[c,d]\to\mathbb R$ continious function. Prove that $g(f_n(x))\to g(f(x))$ Let $\epsilon>0$. If $f_n\to f$, $\exists ...
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 ...
2
votes
1answer
98 views

Regarding Limit/continuity/convergence

let $$f_n(x)=\begin{cases} 1-nx&\text{when }x\in[0,1/n]\\0&\text{when }x\in [1/n,1]\end{cases}$$ Which of the following is correct? $\lim_{ n\to\infty} f_n(x)$ defines a continuous function ...
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: ...
3
votes
1answer
4k views

If $f_n\to f$ uniformly on [a,b] and f is continious on [a,b] then $f_n$ is continious in [a,b]

Yesterday I wrote a test in calculus and had to answer the following question: Prove or contradict: if $f_n\to f$ uniformly on $[a,b]$ and f is continious on [a,b] then $\exists n_0\in\mathbb N$ ...
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
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.
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 $x \in \left[\frac{1}{2} ,1\right]$ ...
1
vote
1answer
120 views

Approximating a function with a polynomials that uniformly converges

Consider f as a continuous function on [a,b]. How can I show that there exists a sequence {$P_n$} of polynomials such that $P_n \rightarrow f$ uniformlyon [a,b] and such that $P_n(a) = f(a)$ for all n ...
1
vote
1answer
284 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. ...
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 ...