Let f(x)=cases(x-[x])x-[x], & x(-2,-1) 2x,,3[|x|], & |x|

Question

Let $f(x)=\begin{cases}\dfrac{\sin(x-[x])}{x-[x]}, & x\in(-2,-1)\ \max\{2x,\,3[|x|]\}, & |x|<1\ 1, & 	ext{otherwise}\end{cases}$ where $[t]$ denotes the greatest integer $\le t$. If $m$ is the number of points where $f$ is not continuous and $n$ is the number of points where $f$ is not differentiable, then the ordered pair $(m,n)$ is:

Accepted Answer

Analysing the pieces and checking the joins at $x=-2,-1,0,1$ (and any interior corner of $\max\{2x,3[|x|]\}$ within $(-1,1)$), the function is found to be discontinuous at exactly $2$ points and non-differentiable at exactly $3$ points, giving $(m,n)=(2,3)$.

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