question_answer

Function\[f(x)=|x|\]is

A)  discontinuous at\[x=0\]

B)  discontinuous at \[x=1\]

C)  continuous at all points

D)  discontinuous at all points

Correct Answer: C

Solution :

 Given, \[f(x)=|x|=\left\{ \begin{matrix}    x, & x>0  \\    0, & x=0  \\    -x, & x<0  \\ \end{matrix} \right.\] \[\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,f(x)=\underset{h\to 0}{\mathop{\lim }}\,|-h|=0,\] \[\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,f(x)=\underset{h\to 0}{\mathop{\lim }}\,|h|=0\] \[\therefore \] \[\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,f(x)=f(0)\] \[\therefore \]At\[x=0,f(x)\]is continuous. Hence, function is continuous at every point. Alternative Method It is clear from the graph that\[f(x)\]is continuous everywhere.