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  • Let a function f(x) be defined by  <img alt="\mathrm{f(x)=\frac{x-|x-1|}{x}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bf%28x%29%3D%5Cfrac%7Bx-%7Cx-1%7C%7D%7Bx

Let a function f(x) be defined by  \mathrm{f(x)=\frac{x-|x-1|}{x}} , then f(x) is  

Option: 1

discontinuous at x=0


Option: 2

discontinuous at x=1


Option: 3

not differentiable at x= -1


Option: 4

not differentiable at x= 2


Answers (1)

best_answer

\mathrm{f(x)=\frac{x-|x-1|}{x}= \begin{cases}\frac{x+x-1}{x}, & x<1, x \neq 0 \\ \frac{x-(x-1)}{x}, & x \geq 1\end{cases}}

\mathrm{= \begin{cases}\frac{2 x-1}{x}, & x<1, x \neq 0 \\ \frac{1}{x}, & x \geq 1\end{cases}}

Clearly f(r) is discontinuous at x=0 as it is not defined at x=0. Since f(x) is not defined at x=0, therefore f(x) cannot be differentiable at x=0. Clearly f(x) is continuous at x=1, but it is not differentiable at x=1, because  \mathrm{L f^{\prime}(1)=1 ~and ~k f^{\prime}(1)=-1}.

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Rishi

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