Let D denote a proper dense subset of a metric space X. Suppose that.f:D→Rf:D\to$$\mathbb{R}$$f:D→R is a uiformly continuous function. For p ∈\in∈ X , Let Bn(p)B_n(p)Bn(p) denote the set {x∈D:d(x,p)<1n}\{x \in D : d(x, p) < $$\frac{1}{n}$$\}{x∈D:d(x,p)<n1}consider : Wp=⋂nf(Bn(p)).W_p = \bigcap_n f(B_n(p)).Wp=n⋂f(Bn(p)).Which of the following statements is true?

WpW_pWp may be empty for some p in X.

WpW_pWp is not empty for every p in X and is contained in f(D).

WpW_pWp is singleton for every p.

WpW_pWp is empty for some p and singleton for some p.

Let D denote a proper dense subset of a metric space X. Suppose that.f:D→Rf:D\to$$\mathbb{R}$$f:D→R is a uiformly continuous function. For p ∈\in∈ X , Let Bn(p)B_n(p)Bn(p) denote the set {x∈D:d(x,p)<1n}\{x \in D : d(x, p) < $$\frac{1}{n}$$\}{x∈D:d(x,p)<n1}consider : Wp=⋂nf(Bn(p)).W_p = \bigcap_n f(B_n(p)).Wp=n⋂f(Bn(p)).Which of the following statements is true?

For Bn(p)={x∈D:d(x,p)<1n}B_n(p) = \\{x \\in D : d(x, p) < $$\\frac{1}{n}$$\\}Bn(p)={x∈D:d(x,p)<n1},we have Wp=⋂nf(Bn(p)).W_p = \\bigcap_n f(B_n(p)).Wp=n⋂f(Bn(p)).By the uniform continuity of f and the denseness of D, Wp=lim⁡n→∞f(Bn(p))={p}.W_p = \\lim_{n \\to \\infty} f(B_n(p)) = \\{p\\}.Wp=n→∞limf(Bn(p))={p}.Thus,Wp W_pWp is a singleton for every p.Correct Answer: (C).$$\\textbf{Correct Answer: (C).}$$Correct Answer: (C).

All Courses

Home

CSIR NET MATHEMATICAL SCIENCES

Real Analysis

Let D denote a proper dense subset of a metric space X. Suppose that.f:D→Rf:D\to\mathbb{R}f:D→R is a uiformly continuous function. For p ∈\in∈&n

Question

Let D denote a proper dense subset of a metric space X. Suppose that.
$f:D\to\mathbb{R}$ is a uiformly continuous function. For p $\in$ X , Let $B_n(p)$ denote the set
$\{x \in D : d(x, p) < \frac{1}{n}\}$
consider : $W_p = \bigcap_n f(B_n(p)).$
Which of the following statements is true?

$W_p$ may be empty for some p in X.

$W_p$ is not empty for every p in X and is contained in f(D).

$W_p$ is singleton for every p.

$W_p$ is empty for some p and singleton for some p.

Solution

Correct option is C

For

$B_n(p) = \{x \in D : d(x, p) < \frac{1}{n}\}$ ,

we have

$W_p = \bigcap_n f(B_n(p)).$

By the uniform continuity of f and the denseness of D,

$W_p = \lim_{n \to \infty} f(B_n(p)) = \{p\}.$

Thus, $W_p$ is a singleton for every p.

$\textbf{Correct Answer: (C).}$