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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability1 Mark
Question
Determine if f defined by $f(x)=\left\{\begin{array}{c}x^2 \sin \frac{1}{x}, \text { if } x \neq 0 \\ 0, \text { if } x=0\end{array}\right.$ is a continuous function?
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Answer

Here, $\mathop {\lim }\limits_{x \to 0} f(x) = \mathop {\lim }\limits_{x \to 0} {x^2}\sin \frac{1}{x} = 0$ x a finite quantity = 0   

$\left[ {\because \sin \frac{1}{x}{\text{lies between - 1 and 1}}} \right]$

Also f(0) = 0 

Since, $\mathop {\lim }\limits_{x \to 0} f\left( x \right) = f\left( 0 \right)$ therefore, the function f is continuous at x = 0. 

Also,when $x\ne0$ ,then f(x) is the product of two continuous functions and hence Continuous.Hence,f(x) is continuous everywhere.

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