The function f(x) = |x| at x = 0 is | vedclass

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(A) The function $f(x) = |x|$ is defined as $f(x) = x$ for $x \ge 0$ and $f(x) = -x$ for $x < 0$.$1$. Continuity at $x = 0$:$\lim_{x 	o 0^-} f(x) = \

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Continuous but non-differentiable

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(A) The function $f(x) = |x|$ is defined as $f(x) = x$ for $x \ge 0$ and $f(x) = -x$ for $x $1$. Continuity at $x = 0$:$\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^-} (-x) = 0$$\lim_{x 	o 0^+} f(x) = \lim_{x 	o 0^+} (x) = 0$$f(0) = |0| = 0$Since $\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^+} f(x) = f(0)$,the function is continuous at $x = 0$.$2$. Differentiability at $x = 0$:Left-hand derivative $(LHD)$ = $\lim_{h 	o 0^-} \frac{f(0+h) - f(0)}{h} = \lim_{h 	o 0^-} \frac{|h| - 0}{h} = \lim_{h 	o 0^-} \frac{-h}{h} = -1$Right-hand derivative $(RHD)$ = $\lim_{h 	o 0^+} \frac{f(0+h) - f(0)}{h} = \lim_{h 	o 0^+} \frac{|h| - 0}{h} = \lim_{h 	o 0^+} \frac{h}{h} = 1$Since $LHD 
eq RHD$,the function is non-differentiable at $x = 0$.Therefore,the function is continuous but non-differentiable.

The function $f(x) = |x|$ at $x = 0$ is

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