Question

Consider the piecewise defined function $\{\begin{array}{cc}\sqrt{-x},& \text{if}x<0\\ 0,& \text{if}0\le x\le 4\\ x-4,& \text{if}x>4\end{array}$.Choose the answer which best describes the continuity of this function-

• A. the function is unbounded and therefore cannot be continuous
• B. the function is right continuous at $x=0$
• C. the function has a removable discontinuity at 0 and 4, but is continuous on the rest of the real line
• D. the function is continuous on the entire real line

Answer 1

The correct option is D the function is continuous on the entire real line

$f\left(x\right)=\{\begin{array}{cc}\sqrt{-x},& \text{if}x<0\\ 0,& \text{if}0\le x\le 4\\ x-4,& \text{if}x>4\end{array}$
Clearly at $x=0$ L.H.L $=$R.H.L $=0$
and $x=4$, L.H.L $=$R.H.L $=0$
Hence function is continuous in entire real line.