Question

The function $f\left(x\right)=\{\begin{array}{cc}sin\frac{\pi x}{2},& x<1\\ [2x-3]x,& x\ge 1\end{array}$ where [.]
denotes the greatest integer function, is

• A. Continuous and differentiable at x $=$ 1
• B. Continuous but not differentiable at x $=$ 1
• C. Discontinuous at x $=$ 1
• D. None of these

Answer 1

The correct option is C Discontinuous at x $=$ 1

$\underset{x\to 1^{-}}{lim}f\left(x\right)=\underset{x\to 1^{-}}{lim}\sin \frac{\pi x}{2}=1$

$\underset{x\to 1^{+}}{lim}f\left(x\right)=\underset{x\to 1^{+}}{lim}[2x-3]x=-1$

RHL$\ne$LHL

$\therefore$Discontinuous at $x=1$