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) $\lim_{x\to 0}\frac{x^2\sin(\frac{1}{x})}{\sin x}=1$

(ii) A real-valued function of three variables which is continuous everywhere is differentiable.

(iii) The function $F:\mathbb{R}^2\to\mathbb{R}^2,\text{defined by}F(x,y)=(y+2,x+y),$ is locally invertible at any $\left(x,y\right)\epsilon R^{2}$

(iv) $f:[-1,1]\times[-2,2]\to\mathbb{R},\text{defined by}$

$f(x,y)=\begin{cases}x,&\text{if}y\text{is rational}\\0,&\text{if}y\text{is not rational}\end{cases}$ is integrable.

(v) The function $f:\mathbb{R}^2\to\mathbb{R},\text{defined by}f(x,y)=1-y^2+x^2,$ (.0,0)

17 May 2025

Answer :

Word Count : 153

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