Question

Discuss the continuity and differentiability of the function $f\left(x\right)=\left|x\right|+\left|x-1\right|$ in the interval $\left(-1,2\right).$

Answer 1

$f\left(x\right)=|x|+|x-1|$

It can be written as

$f\left(x\right)=\{\begin{array}{cc}-2x+1;& x<0\\ +1;& 0\le x<1\\ 2x-1& x\ge 1\end{array}$

At $x=0$

$LHL=-2\left(0\right)+1=1$

$RHL=1$

$LHD=-2$

$RHD=0$

Continuous but not differentiable on $x=0$

At $x=1$ $LHL=2\left(1\right)-1=1$

$RHL=1$

$LHD=1$

$RHD=2$

Continuous but not differentiable on $x=1$

Continuous everywhere in $(-1,2)$ but not differentiable on $x=0,1$ in $(-1,2)$.