State whether the following statements are true or false. Give reasons

Question

State whether the following statements are true or false. Give reasons for your answers.
(i) $\displaystyle \lim_{x \to 0} \frac{x^2 \sin \frac{1}{x}}{\sin x} = 1$
(ii) A real-valued function of three variables which is continuous everywhere is differentiable.
(iii) The function $F : \mathbb{R}^2 \to \mathbb{R}^2$ , defined by $F(x, y) = (y + 2, x + y)$ , is locally invertible at any $(x, y) \in \mathbb{R}^2$ .
(iv) $f : [-1,1] \times [-2,2] \to \mathbb{R}$ , defined by
$$f(x, y) = \begin{cases} x, & \text{if } y \text{ is rational} \\ 0, & \text{if } y \text{ is not rational} \end{cases}$
is integrable.
(v) The function $f : \mathbb{R}^2 \to \mathbb{R}$ , defined by $f(x, y) = 1 - y^2 + x^2$ , has an extremum at (0, 0).

09 Jan 2026

Answer :

Word Count : 141

(i) False. Not all equations automatically satisfy the conditions for the given context without specifying the exact form; assumptions about differentiability, continuity, or constraints are necessary to claim truth. (ii) False. Continuity everywhere does not ________ ___ _______ _____ _______.
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