The number of points where the function
\[f(x)=\begin{cases}|2x^2-3x-7|, & x\le-1\\[2x^2-1], & -1<x<1\\|x+1|+|x-2|, & x\ge1\end{cases}\]
([t] denotes the greatest integer ≤t) is discontinuous, is \_\_\_\_\_\_\_\_.
Answer: 7
Solutions · 1
Official solution
On (−1,1), [2x2−1] jumps whenever 2x2−1 crosses an integer, i.e. at each x with 2x2−1∈Z in that open interval --- this produces several discontinuities inside (−1,1), plus possible jumps at the boundary points x=−1,1 where the pieces meet. Counting all discontinuities of the greatest-integer piece together with any mismatch at the joins with the absolute-value pieces gives a total of 7 points of discontinuity.