mathematics question

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Define the function

$f (x) = {\begin{matrix} x^{2} \sin ? (\frac{1}{x^{2}}), & x 0, \\ 0, & x = 0. \end{matrix}$ $f (x) = x^{2} sin (x ^{2} 1), 0, x = 0, x = 0.$

(a)

Prove that $f (x)$ is continuous at $x = 0$ .

(b)

Determine whether

$f (x)$ $f (x)$ is differentiable at $x = 0$ . If so, find $f^{} (0)$ .

(c)

Find an explicit formula for $f^{} (x)$ for $x 0$ .

(d)

Decide whether

$f^{} (x)$ $f (x)$ is bounded on any neighborhood of $x = 0$ .

(e)

Determine whether

$f^{} (x)$ $f (x)$ is Riemann integrable on $[1, 1]$ .

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mathematics question

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