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(A) By definition,the absolute value function is given by:$f(x) = \begin{cases} -x, & 	ext{if } x < 0 \ x, & 	ext{if } x \ge 0 \end{cases}$Clearly,

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The function is continuous at $x = 0$.

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(A) By definition,the absolute value function is given by:$f(x) = \begin{cases} -x, & 	ext{if } x Clearly,the function is defined at $x = 0$ and $f(0) = 0$.The left-hand limit $(LHL)$ of $f$ at $x = 0$ is:$\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^-} (-x) = 0$The right-hand limit $(RHL)$ of $f$ at $x = 0$ is:$\lim_{x 	o 0^+} f(x) = \lim_{x 	o 0^+} (x) = 0$Since $\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^+} f(x) = f(0) = 0$,the function $f$ is continuous at $x = 0$.

Discuss the continuity of the function $f$ given by $f(x) = |x|$ at $x = 0$.

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