Question

Let $\left[t\right]$ denote the greatest integer less than or equal to $t.$ Let $f\left(x\right)=x-\left[x\right],g\left(x\right)=1-x+\left[x\right],$ and $h\left(x\right)=min\left\{f\right(x),g(x\left)\right\},x\in [-2,2].$ Then $h$ is

• A. not continuous at exactly four points in $[-2,2]$
• B. continuous in $[-2,2]$ but not differentiable at more than four points in $(-2,2)$
• C. not continuous at exactly three points in $[-2,2]$
• D. continuous in $[-2,2]$ but not differentiable at exactly three points in $(-2,2)$

Answer 1

The correct option is B continuous in $[-2,2]$ but not differentiable at more than four points in $(-2,2)$

$f\left(x\right)=\left\{x\right\}\text{and}g\left(x\right)=1-\left\{x\right\}$
$h\left(x\right)=min\left\{f\right(x),g(x\left)\right\}$


$\therefore h\left(x\right)$ is continuous everywhere and non-diffeerentiable at 7 points.