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(A) Given the function $f(x) = \frac{x}{1 + |x|}$.We analyze the differentiability of $f(x)$ at $x = 0$ and other points.For $x > 0$,$f(x) = \frac{x}{

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$( - \infty, \infty )$

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(A) Given the function $f(x) = \frac{x}{1 + |x|}$.We analyze the differentiability of $f(x)$ at $x = 0$ and other points.For $x > 0$,$f(x) = \frac{x}{1 + x}$,which is a rational function with a non-zero denominator,hence differentiable.For $x Now,check differentiability at $x = 0$ using the definition of the derivative:$f'(0) = \lim_{h 	o 0} \frac{f(h) - f(0)}{h - 0} = \lim_{h 	o 0} \frac{\frac{h}{1 + |h|} - 0}{h} = \lim_{h 	o 0} \frac{1}{1 + |h|}$.As $h 	o 0$,$|h| 	o 0$,so the limit is $\frac{1}{1 + 0} = 1$.Since the limit exists and is finite,the function $f(x)$ is differentiable at $x = 0$.Thus,the function is differentiable for all $x \in ( - \infty, \infty )$.

The set of all those points,where the function $f(x) = \frac{x}{1 + |x|}$ is differentiable,is

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