Uniform convergence Definition and 164 Threads

Here is the question from the book: ---------- Let f: \mathbb{R} \to \mathbb{R} be a function. For any a\in \mathbb{R}, let f_a :\mathbb{R}\to \mathbb{R} be the shifted function f_a(x):=f(x-a).(a) Show that f is continuous if and only if, whenever (a_n)_{n=0}^{\infty} is a sequence of real...

Suppose f(x) is continuous on [0,1], and that f(1)=0. Prove that x^n f(x) converges uniformly on [0,1] as n \rightarrow \infty By continuity, if |x-1|< \delta then |f(x)|< \epsilon for x \in [x_0 ,1] for some x_0 \in [0,1]. And there is some N such that if n>N, then |x^n|<\epsilon since...

Hi, I'm studying for finals and I just need some feedback. One of questions MIGHT be: If I know a sequence of functions is pointwise convergent, how do I show that it's not uniformly convergent? I think that a pointwise convergent sequence of functions might not converge to a continuous...

This question arised in my last math class: If a sequence of functions f_n uniformly converges to some f on (a, b) (bounded) and all f_n are integrable on (a, b), does this imply that f is also integrable on (a, b) ?? (f_n do not necessarily have to be continous, if they were, the answer...

"Suppose f_n are defined and continuous on an interval I. Assume that f_n converges uniformly to f on I. If a_n in I is a sequence and a_n -> a, prove that f_n(a_n) converges to f(a)." I don't understand the question. Doesn't uniform convergence imply that for all x in I and e>0, | f_n(x) -...

Hello, I have two questions to ask regarding uniform convergence for sequences of functions. So I know that if a sequence of continuous functions f_n : [a,b] -> R converge uniformly to function f, then f is continuous. Is this true if "continous" is replaced with "piecewise continuous"...

does Fn(x)= nx(1 - x^2)^n converge uniformly on [0,1]? my first instinct was yes it converges uniformly to 0 but I can't seem to show that using the definition. i get |nx(1 - x^2)^n|<=|nx|<=n for x in [0,1] any tip or hint would be helpful thanks

Discuss the uniform convergence of the following sequence in the interval indicated {x^n} , 0< x <1 Now, f(x) = \lim_{n\rightarrow \infty} f_{n}(x) = 0 Therefore given any small \epsilon > 0 , if there exists N such that |f_n(x)-f(x)| < \epsilon for all n \geq N for all x in the...

I haven't done uniform convergence since last year when I took analysis, and now I have this problem for topology (we're studying metric spaces right now) and I can't remember how to disprove uniform convergence: f_n: [0,1] -> R , f_n(x)=x^n Show the sequence f_n(x) converges for all x in...

Suppose (f_n} is a sequence of functions where f_n(x) = x / (1 + n^2 x^2). I am finding the pointwise limit of the sequence of {f_n'(x)} on the interval (-oo, + oo)...in which {f_n'(x)} is the sequence of functions obtained from the derivative of x / (1 + n^2 x^2) and I am trying to find...

Hi, I don't know how to analyse uniform convergence/local uniform convergence for this series of functions: \sum_{n=1}^{\infty}2^{n}\sin \frac{1}{3^{n}x} Then f_{n}^{'} = -\left(\frac{2}{3}\right)^{n}\frac{\cos \frac{1}{3^{n}x}}{x^2} f_{n}^{'} = 0 \Leftrightarrow x =...

Hi, I've little probles with this one: Analyse uniform and local uniform convergence of this series of functions: \sum_{k = 1}^{\infty} \frac{\cos kx}{k} I'm trying to solve it using Weierstrass' criterion, ie. \mbox{Let } f_n \mbox{ are defined on } 0 \neq M \subset...

Hi, next one I've little problems with: f_{n} = \frac{2nx}{1+n^{2}x^{2}} \mbox{a) } x \in [0, 1] \mbox{b) } x \in (1, \infty) First the pointwise convergence: \lim_{n \rightarrow \infty} \frac{2nx}{1+n^{2}x^{2}} = 0 I computed the derivative of my function to...

Hi, let's suppose f_{n}(x) = \frac{x^{n}}{1+x^{n}}, a) x \in [ 0, 1 - \epsilon] b) x \in [ 1 - \epsilon, 1 + \epsilon] c) x \in [ 1 + \epsilon, \infty] Where \epsilon \in \left( 0, 1 \right). Analyse pointwise, uniform and locally uniform convergence. Well, we...