Question

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

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

Answer 1

Answer: (A)

$\min \{x-[x], 1-x+[x]\}$
$h(x)=\min \{x-[x], 1-[x-[x])\}$

$\Rightarrow $ always continuous in $[-2,2]$
but non differentiable at $7$ Points