Linked Questions

5
votes
1answer
6k views

Limit at infinity of a uniformly continuous integrable function [duplicate]

Possible Duplicate: $f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$ This is an exercise from Berkeley preliminary exams, Fall 1983 Let $ f : [...
2
votes
2answers
1k views

Uniformly continuous function $\rightarrow 0$ as $x\rightarrow\infty$ [duplicate]

Possible Duplicate: $f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$ Given that $f$ is uniformly continuous function on real and is integrable ...
1
vote
2answers
374 views

Prove $f$ is uniformly continuous iff $ \lim_{x\to \infty}f(x)=0$ [duplicate]

Let $f:[0,\infty)\to (0,\infty)$ be a continuous function and $\displaystyle\int^{\infty}_{0}f<\infty$. Show that $f$ is uniform continuous iff $\displaystyle\lim_{x\to \infty}f(x)=0$ So I could ...
1
vote
0answers
118 views

uniformly continuous and $\int_0^\infty f(t)\,\mathrm dt$ exists $\implies \lim_{x\to\infty}f(x) = 0 $ [duplicate]

I appreciate your help with this one. Let $f \colon[0,\infty)\rightarrow \mathbb{R}$ be uniformly continuous and let the integral $\int_0^\infty f(t)\,\mathrm dt$ exist and be final. I need to show ...
0
votes
0answers
65 views

Uniformly continuous function which is integrable but does not have a limit [duplicate]

Is there an example of a function $f:[0,+\infty)\to \mathbb{R}$ which is uniformly continuous and $\int_0^{+\infty}|f(x)|dx<+\infty$, but $\lim_{x\to+\infty}f(x)\neq0$ (since it is integrable this ...
1
vote
0answers
34 views

f uniformly Lipschitz, integral from 0 to inf of |f(x)| < inf imply |f(x)| converges to 0 as x -> inf. [duplicate]

Suppose f: [0,inf) -> R is uniformly Lipschitz and the integral from 0 to inf of |f(x)| < inf. Then|f(x)| converges to 0 as x -> inf. Prove this by proving the contrapositive. ie. negate the limit ...
3
votes
1answer
3k views

Does an absolutely integrable function tend to $0$ as its argument tends to infinity?

Suppose that $f:[0,\infty)\rightarrow\mathbb{R}$ is continuous. Is it true that $$\int_{0}^\infty|f(t)|dt<\infty\Rightarrow \lim_{t\rightarrow\infty}f(t)=0?$$ If so can you provide a proof, ...
2
votes
3answers
139 views

Is the sequence $\big(f(n)-f(n+1)\big)$ convergent?

Let $f:\mathbb{R}\to[0,+\infty)$ be a function such that $\int_{-‎\infty‎}^{+\infty}f(x)\,dx=1$. My question is: Is the sequence $\big(f(n)-f(n+1)\big)$ convergent? I found that there exist some $f$...
2
votes
1answer
540 views

If $f:\mathbb{R}\to[0, \infty)$ (uniformly) continuous and $f \in L^1$, then $\lim_{x\to\pm\infty}f(x)=0$?

I'm learning about measure theory and need help with the following questions: True or False (justify): $(1)$ If $f:\mathbb{R}\to[0, \infty)$ measurable and $f \in L^1$, then $\lim_{x\to\pm\...
0
votes
1answer
930 views

if f is continuous and absolutely integrable $\lim\limits_{x\to\infty}f(x)=0$

Prove that if f(x) is continuous and absolutely integrable on $[a,\infty)$ then $\lim\limits_{x\to\infty}f(x)=0$. I tried proving it in the following way: First we need to prove the existence of the ...
2
votes
2answers
275 views

Assume that $f$ is uniformly continuous. Prove that $\lim_{x→∞} f(x) = 0.$

I think the following question is probably fairly easy but can't think of an easy way of proving it. Some help would be awesome. This question comes from an old qual. Thanks. Let $f$ be an ...
11
votes
1answer
304 views

Essential Selfadjointness of Quantum Harmonic Oscillator Hamiltonian

The Hamiltonian for the Quantum Harmonic Oscillator is (disregarding constants) the Hermite operator $$ Hf = -f''+x^{2}f, $$ where $\mathcal{D}(H)$ consists of all twice absolutely ...
0
votes
0answers
451 views

Behaviour of $L^2$ functions at infinity [duplicate]

Is it possible to prove that if $f\in L^2(\mathbb {R}) $ then $\exists\lim_{x\to\pm\infty}\lvert f\rvert^2$ and $\lim_{x\to\pm\infty}\lvert f\rvert^2=0$? If not, is it easy to find a counterexample?
3
votes
2answers
106 views

When does $f\in L^1$ vanish?

I am asking about the sort-of converse to this question: under what additional conditions on $f:\mathbb{R}_+\rightarrow\mathbb{R}$ does the following hold? $$ \int_{\mathbb{R}_+}f<\infty\implies \...
2
votes
2answers
81 views

Suppose $f : X \to Y$ is a (continuous) bounded map.Does this implies that $f$ is uniformly continuous?

It's well known that if $ f : \bf (X,d) \to \bf (Y,e) $ is a uniformly continuous function then $f$ maps bounded set to bounded set.Does the converse hold ? More Precisely, Suppose $f : X \to Y$ ...

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