Questions tagged [sequence-of-function]

Use this tag only when your query is about sequences of functions. Don't use this tag for any other sequence such as sequences of real numbers or sequences of complex numbers etc.

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3
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1answer
65 views

Uniform convergence on two separate intervals

Test the sequence of functions for uniform convergence $$f_n(x)=\frac{nx}{1+n^2x^2}$$ $a) \text{on} \ [0,1]$ $b) \text{on} \ [1,{\infty})$ a) If we differentiate this with respect to ...
2
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1answer
302 views

Showing that $f_n(x) = \frac{nx}{1+n^2x^2}$ does not converge uniformly

I am trying to show that $f_n(x) = \frac{nx}{1+n^2x^2}$ where $x\in \mathbb{R}$ does not converge uniformly. I have made an attempt and would like to make sure that I am going about it in the right ...
-1
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1answer
46 views

Constructing a sequence of function [closed]

Construct a sequence of functions on [0,1] each of which is discontinuous at every point on [0,1] and which converges uniformly to a function that is continuous at every point?
3
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1answer
813 views

Given sequence of $L-$Lipschitz functions which converges pointwise, prove uniform convergence

Let $f_n:[a.b]\rightarrow \mathbb{R}$ be sequence of $L-$Lipschitz functions, that is: $$\forall x,y\in[a,b]: |f_n(x)-f_n(y)|\leq L|x-y|$$ Suppose $f_n \rightarrow f$ pointwise, prove $f_n \...
2
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1answer
79 views

Convergence to Riemann-Stieltjes integral of sequence of Riemann-Stieltjes-like sums with changing integrand and integrator

I am considering the limiting behavior of a sequence of Riemann-Stieltjes (RS) (or at least RS-like) sums in the sense of their convergence to a Riemann-Stieltjes integral. The general term has the ...
1
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1answer
70 views

Let $\{f_n\}$ be a sequence of unif cont functions from $(X,d) \to (\Omega,\rho)$ s.t $\{f_n\} \to f$ uniformly. Show that $f$ is unif cont.

This is what I have so far: $\textbf{Proof:}$ We must show that $f$ is uniformly continuous, hence satisfy that $\forall \epsilon > 0, \exists \delta > 0, \forall x,y \in X$ with $d(x,y) < ...
1
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1answer
39 views

Limiting total variation attached to sequence of uniformly vanishing functions of bounded variation

Let $(f_n)_n$ be a sequence of real functions of a single real variable with compact support in $[0,1]$ and of bounded variation all of them. Let the sequence be uniformly convergent to $0$. Is it ...
1
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1answer
35 views

continuity of the function $f(x)=\lim_{n\to \infty}\sum_{k=0}^{n-1} \dfrac{x}{(kx+1)[(k+1)x+1]}$

I need to check the continuity and differentiability of the function $f(x)$ at $x=0$ where, $$f(x)=\lim_{n\to \infty}\sum_{k=0}^{n-1} \dfrac{x}{(kx+1)[(k+1)x+1]}.$$ I tried to check the domain of ...
1
vote
2answers
400 views

Show that the sum of reciprocal products equals $n$

I don't even know how to proceed. Please help me with this. (Original at https://i.stack.imgur.com/DRIX8.jpg) Consider all non-empty subsets of the set $\{1, 2, \ldots, n\}$. For every such ...
0
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1answer
37 views

Sequence of functions uniformly differentiable at a point?

Let $\{f_{n}\}$ be a sequence of functions differentiable at $x_0$. Let $r_{n}(h,x_0) = f_n(x_0+h) - f_n(x_0) - f'_n(x_0)h $ We know that for any fixed $n$, $r_n(h,x_0) = o(|h|)$ as $h \to 0$ from ...