Let [x] denote the greatest integer function, and let m and n respectively be the numbers …

Question

Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x)=[x]+|x-2|,-2< x<3$, is not continuous and not differentiable. Then $\mathrm{m}+\mathrm{n}$ is equal to :

Accepted Answer

On each unit sub-interval, $[x]$ is constant, so:
$$f(x)=\begin{cases}-x, & -2<x<-1\ -x+1, & -1\le x<0\ -x+2, & 0\le x<1\ -x+3, & 1\le x<2\ x, & 2\le x<3\end{cases}$$

$[x]$ jumps at each integer in $(-2,3)$: $x=-1,0,1,2$, so $f$ is discontinuous at these $4$ points ($m=4$), and being discontinuous there it is automatically non-differentiable there too, so $n=4$.

$m+n=4+4=8$

Solutions · 1