If f(x) = begin{cases} x, & x le 0 0, & x | vedclass

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(C) Given $f(x) = \begin{cases} x, & x \le 0 \ 0, & x > 0 \end{cases}$.For continuity at $x = 0$:Left Hand Limit $(LHL)$ = $\lim_{x 	o 0^-} f(x) = \

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continuous but not differentiable

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(C) Given $f(x) = \begin{cases} x, & x \le 0 \ 0, & x > 0 \end{cases}$.For continuity at $x = 0$:Left Hand Limit $(LHL)$ = $\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^-} x = 0$.Right Hand Limit $(RHL)$ = $\lim_{x 	o 0^+} f(x) = \lim_{x 	o 0^+} 0 = 0$.Value of function $f(0) = 0$.Since $LHL$ = $RHL$ = $f(0)$,the function is continuous at $x = 0$.For differentiability at $x = 0$:Left Hand Derivative $(LHD)$ = $\lim_{h 	o 0^+} \frac{f(0-h) - f(0)}{-h} = \lim_{h 	o 0^+} \frac{(0-h) - 0}{-h} = \lim_{h 	o 0^+} \frac{-h}{-h} = 1$.Right Hand Derivative $(RHD)$ = $\lim_{h 	o 0^+} \frac{f(0+h) - f(0)}{h} = \lim_{h 	o 0^+} \frac{0 - 0}{h} = 0$.Since $LHD$ $
eq$ $RHD$,the function is not differentiable at $x = 0$.

If $f(x) = \begin{cases} x, & x \le 0 \\ 0, & x > 0 \end{cases}$ then the function $f(x)$ at $x = 0$ is:

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