PG entrance exam IIT JAM 2025 Mock Test Mathematical Science Analysis Continuity & Differentiability
Let \(D \subset \mathbb{R}^2 \) be defined by \(D = \mathbb{R}^2 \setminus \{(x, 0) : x \in \mathbb{R}\} \) . Consider the function\( f : D \to \mathbb{R} \) defined by
\(f(x, y) = \frac{x \sin \left(\frac{1}{y}\right)}{y}. \)
Which of the following is true?
1
f is a discontinuous function on D .
2
f is a continuous function on D and cannot be extended continuously to any point outside D .
3
f is a continuous function on D and can be extended continuously to \( D \cup \{(0, 0)\} \).
4
f is a continuous function on D and can be extended continuously to the whole of \( \mathbb{R}^2 \) .