Teaching Haryana (HPSC) Assistant Professor Mock Test 2025 Mathematical Science Analysis Continuity & Differentiability
Let D denote a proper dense subset of a metric space X. Suppose that f : D → ℝ is a uniformly continuous function. For p ∈ X, let Bn(p) denote the set
\(\left\{x \in D: d(x, p)<\frac{1}{n}\right\} \)
Consider \(\left.W_p=\bigcap_n \overline{f\left(B_n(p)\right.}\right) \).
Which of the following statements is true?
1
Wp may be empty for some p in X.
2
Wp is not empty for every p in X and is contained in f(D).
3
Wp is a singleton for every p.
4
Wp is empty for some p and singleton for some p.
5
Question Not Attempted