Question

# Consider the following in respect of the function $$f(x) = | x- 3 |$$ :1. $$f(x)$$ is continuous at $$x = 3$$2. $$f(x)$$ is differentiable at $$x = 0$$.Which of the above statements is/are correct ?

A
1 only
B
2 only
C
Both 1 and 2
D
Neither 1 nor 2

Solution

## The correct option is C Both 1 and 2$$\displaystyle \lim _{ x\rightarrow { 3 }^{ - } }{ f\left( x \right) } =\displaystyle \lim _{ x\rightarrow { 3 }^{ - } }{ -\left( x-3 \right) } =-\left( 3-3 \right) =0$$$$\displaystyle \lim _{ x\rightarrow { 3 }^{ + } }{ f\left( x \right) } =\displaystyle \lim _{ x\rightarrow { 3 }^{ + } }{ \left( x-3 \right) } =3-3=0$$$$f\left( 3 \right) =\left| 3-3 \right| =0$$LHL$$=$$RHL $$=f\left( x \right) \Rightarrow$$continuous at $$x=3$$$$f'\left( x \right) =\begin{cases} -1;\quad x<3 \\ 0;\quad x=3 \\ 1;\quad x>3 \end{cases}$$$$f'\left( x \right) =-1$$ for $$x<3$$$$\Rightarrow f'\left( x \right) =-1$$ for $$x=0$$$$\Rightarrow f\left( x \right)$$ is differentiable at $$x=0$$Mathematics

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