Identity function, modulus and signum for JEE

Formula

\[I(x)=x,\qquad |x|=\begin{cases}x,&x\ge 0\\-x,&x<0\end{cases},\qquad \operatorname{sgn}(x)=\begin{cases}1,&x>0\\0,&x=0\\-1,&x<0\end{cases}\]

Variables: \(|x|\) is modulus and \(\operatorname{sgn}(x)\) is signum.

Piecewise continuity/differentiability, graph-based questions, derivative of modulus forms.