Question

Let [t] denote the greatest integer \(\le t\). The number of points where the function \(f(x)=[x]\left|x^2-1\right|+\sin\left(\frac{\pi}{[x]+3}\right)-[x+1],\ x\in(-2,2)\) is not continuous is _____________.

Hint:

Only the integer points \(x=-1,0,1\) can be points of discontinuity; compare left limit and value at each.

Answer 1

Correct Answer: 2

Step 1: On each open interval between integers, [x] is constant, so f is continuous there. Check \(x=-1,0,1\).
Step 2: Note \(\sin\frac{\pi}{[x]+3}\) equals \(0,1,\frac{\sqrt3}{2},\frac{\sqrt2}{2}\) for \([x]=-2,-1,0,1\), and \([x+1]=[x]+1\).
Step 3: At \(x=-1\): left limit \(=(-2)(0)+0-(-1)=1\); value \(=(-1)(0)+1-0=1\). Continuous.
Step 4: At \(x=0\): left limit \(=(-1)(1)+1-0=0\); value \(=0+\frac{\sqrt3}{2}-1\approx-0.13\). Not continuous.
Step 5: At \(x=1\): left limit \(=0+\frac{\sqrt3}{2}-1\approx-0.13\); value \(=1\cdot0+\frac{\sqrt2}{2}-2\approx-1.29\). Not continuous.
Hence f is discontinuous at 2 points.
Answer: 2