List of top Mathematics Questions on Continuity

Let \(f(x, y) = \begin{cases} \frac{xy}{\sqrt{x^2 + y^2}}, & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases}\), then

Which of the following statements are correct:
A. The set of limit points of the set of rationals $\mathbb{Q}$ is empty in the real line $\mathbb{R}$. B. $(0,1)$ is open in the real line $\mathbb{R}$. C. The set $\{\frac{1}{n} \mid n \in \mathbb{N}\} \cup \{0\}$ is compact in $\mathbb{R}$. D. The set $\{x : |x| > 1\}$ is a connected subset of $\mathbb{R}$. E. $[0,1]$ is a closed and compact subset of $\mathbb{R}$. Choose the correct answer from the options given below:

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : The set of rationals $\mathbb{Q}$ is closed in the real line $\mathbb{R}$. Reason R : The closure of $\mathbb{Q}$ in $\mathbb{R}$ is $\mathbb{R}$. In the light of the above statements, choose the correct answer from the options given below

Which of the following set is open in the real line $\mathbb{R}$?

Which of the following is a disconnected subset of the real line $\mathbb{R}$?

Prove that the function \( f(x) = |x| \), is continuous at \( x = 0 \).

Prove that the function \[ f(x) = \begin{cases} x^3 - 3 & \text{if } x \leq 2 \\ x^2 + 1 & \text{if } x>2 \end{cases} \] is continuous at \( x = 2 \).