Question

Show that $f\left(x\right)=|(1+x)+|x||$ is continuous for all $x\in R$.

Answer 1

For $x>0$

The value of the function.

$f\left(x\right)=|1+x+|x||=|1+2x|=1+2x$

The left hand limit.

$LHL=\underset{h\to 0}{Lt}|1+x+h+|x+h||=1+2x$

The right hand limit.

$LHL=\underset{h\to 0}{Lt}|1+x-h+|x-h||=1+2x$

Thus the function is continuous for $x>0$

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

For $x<0$

The value of the function.

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

The left hand limit.

$LHL=\underset{h\to 0}{Lt}|1+x+h+|x+h||=1+x+h-x-h=1$

The right hand limit.

$LHL=\underset{h\to 0}{Lt}|1+x-h+|x-h||=1+x-h-x+h=1$

Thus the function is continuous for $x<0$

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