Question

If $f\left(x\right)=\text{sgn}\left(x\right)$ and $g\left(x\right)=x(x^2-5x+6)$, then $f\left(g\right(x\left)\right)$ will be discontinuous at

• A. infinitely many points
• B. exactly one point
• C. exactly three points
• D. no point

Answer 1

The correct option is C exactly three points

Given : $g\left(x\right)=x(x^2-5x+6)$
The roots of $g\left(x\right)$ are $0,2,3$
Hence $f\left(g\right(x\left)\right)$ would change sign at these three points. Hence it is discontinuous at these $3$ points
Note: $\text{sgn}\left(g\right(x\left)\right)$ is discontinuous when $g\left(x\right)=0$