GATE 2017 EE (EE2) – Question 36
Let
$$g(x)=\begin{cases}-x,&x\le1,\\x+1,&x\ge1,\end{cases}$$
and
$$f(x)=\begin{cases}1-x,&x\le0,\\x^2,&x>0.\end{cases}$$
The number of discontinuities of $(f\circ g)(x)=f(g(x))$ on $(-\infty,0)$ is
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: (A) 0
Explanation
Only negative x are relevant. Every such x satisfies x≤1, so g(x)=−x>0 and the positive branch of f applies.
Therefore $f(g(x))=(-x)^2=x^2$ throughout the requested open interval. This is continuous there, giving **zero discontinuities, A**.
The source assigns overlapping cases at x=1 in g; that point lies outside the interval and does not affect this answer.