Let $$f\left( x \right) = - 1 + \left| {x - 2} \right|,$$     and $$g\left( x \right) = 1 - \left| x \right|;$$    then the set of all points where $$fog$$  is discontinuous is :

Solution :
\[\begin{array}{l} f\left( {g\left( x \right)} \right) = f\left( {1 - \left| x \right|} \right) = - 1 + \left| {\left| x \right| + 1} \right|\\ {\rm{Let\,\, }}fog = y\\ \therefore \,y = - 1 + \left| {\left| x \right| + 1} \right| \Rightarrow y\left\{ \begin{array}{l} \,\,\,x,\,\,\,\,x \ge 0\\ - x,\,\,\,\,x < 0 \end{array} \right. \end{array}\]
$$\eqalign{ & {\text{L}}{\text{.H}}{\text{.L}}{\text{. at }}\left( {x = 0} \right) = \mathop {\lim }\limits_{x \to 0} \left( { - x} \right) = 0 \cr & {\text{R}}{\text{.H}}{\text{.L}}{\text{. at }}\left( {x = 0} \right) = \mathop {\lim }\limits_{x \to 0} \left( x \right) = 0 \cr & {\text{When }}x = 0,{\text{ then }}y = 0 \cr & {\text{Hence, L}}{\text{.H}}{\text{.L}}{\text{. at }}\left( {x = 0} \right) = {\text{ R}}{\text{.H}}{\text{.L at }}\left( {x = 0} \right) = {\text{value of }}y{\text{ at }}\left( {x = 0} \right) \cr} $$
Hence, $$y$$ is continuous at $$x = 0$$
Clearly at all other point $$y$$ continuous. Therefore, the set of all points where $$fog$$  is discontinuous is an empty set.