Question

If $f\left(x\right)=\left|\begin{array}{c}{\log }_{10}x\end{array}\right|$ then at $x=1$

• A. $f$ is not continuous
• B. $f$ is continuous but not differentiable
• C. $f$ is differentiable
• D. the derivative is 1

Answer 1

The correct option is B $f$ is continuous but not differentiable

$f\left(x\right)=\left|{\log }_{10}x\right|$

We know that $\log x>0$ for $x>1$

and $\log x<0$ for $0<x<1$

$\log x=0$ for $x=1$

${lim}_{x\to 1^{+}}f\left(x\right)={lim}_{x\to 1^{+}}\left(\log x\right)=\log 1=0$

${lim}_{x\to 1^{-}}f\left(x\right)={lim}_{x\to 1^{-}}(-\log x)=-\log 1=0$

$f\left(1\right)=0$

LHL$=$RHL $=f\left(x\right)$

$\therefore$ It is continuous at $x=0$

$f\left(x\right)=\{\begin{array}{c}\log xx>1\\ 0x=1\\ -\log x0<x<1\end{array}$

$f^{\prime }\left(x\right)=\{\begin{array}{c}1/xx>1\\ 0x=1\\ -1/x0<x<1\end{array}$

${lim}_{x\to 1^{+}}{f}^{\prime }\left(x\right)=\frac{1}{1}=1$

${lim}_{x\to 1^{-}}{f}^{\prime }\left(x\right)=\frac{-1}{1}=-1$

LHL$\ne$RHL

$\therefore$ It is not differentiable at $x=1$

For $f\left(x\right)$ to be continuous

RHL$=f\left(x\right)$ at $x=0$

$\Rightarrow \frac{1}{12}=f\left(0\right)$