Question

Let f(x) = x - |x| then f(x) is,

• A. Continuous ⩝×∈ R
• B. discontinuous at x = 0
• C. Always increasing
• D. a constant

Answer 1

The correct option is A

Continuous ⩝×∈ R

Given function is,
$f\left(x\right)=x-|x|$
when $x>0$
$f\left(x\right)=x-x=0$
when $x<0$
$f\left(x\right)=x-(-x)$
$=2x$

$\therefore$ The graph for the above can be represented as below.

This shows that no breakage is happening at any point including at x = 0. That is the function is continuous at every point x $ϵ$ R.

Also its not strictly increasing since in $x>$ 0 the function is constant, that is nor then increasing nor decreasing.