The function f(x) = x + |x| is continuous for | vedclass

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(A) The given function is $f(x) = x + |x|$.We can define the absolute value function $|x|$ as:$|x| = \begin{cases} x, & x \geq 0 \ -x, & x < 0 \end{c

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$x \in (-\infty, \infty)$

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(A) The given function is $f(x) = x + |x|$.We can define the absolute value function $|x|$ as:$|x| = \begin{cases} x, & x \geq 0 \ -x, & x Therefore, the function $f(x)$ can be written as:$f(x) = \begin{cases} x + x, & x \geq 0 \ x - x, & x Now, we check the continuity at $x = 0$:Left-hand limit: $\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^-} (0) = 0$Right-hand limit: $\lim_{x 	o 0^+} f(x) = \lim_{x 	o 0^+} (2x) = 2(0) = 0$Value of the function: $f(0) = 2(0) = 0$Since $\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^+} f(x) = f(0) = 0$, the function is continuous at $x = 0$.Since $f(x) = 2x$ is a polynomial for $x > 0$ and $f(x) = 0$ is a constant function for $x < 0$, the function is continuous for all $x \in (-\infty, \infty)$.

The function $f(x) = x + |x|$ is continuous for:

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