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

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(A) Continuity at $x = 0$: $\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0} x = 0$ $\lim_{x 	o 0^+} f(x) = 0$ $f(0) = 0$ Since $\lim_{x 	o 0^-} f(x) = \lim_{

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Continuous but not differentiable

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(A) Continuity at $x = 0$: $\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0} x = 0$ $\lim_{x 	o 0^+} f(x) = 0$ $f(0) = 0$ Since $\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^+} f(x) = f(0)$,the function is continuous at $x = 0$. Differentiability at $x = 0$: Left-hand derivative $(LHD)$ = $\lim_{h 	o 0^+} \frac{f(0-h) - f(0)}{-h} = \lim_{h 	o 0^+} \frac{-h - 0}{-h} = 1$ 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 $x = 0$. Conclusion: The function is continuous but not differentiable at $x = 0$.

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

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