Let [x] be the greatest integer ≤ x. Then the number of points in the interval (-2,1), where the function f(x)=|[x]|+√(x-[x]) is discontinuous, is ___________.

Solution

Maths sir · 3 steps

The only suspects are the integers inside (-2,1): check -1 and 0 with a left limit and a value.

1. Where to look. Between two integers, [x] stays fixed and x-[x] changes smoothly, so f is continuous there. Only the integers -1 and 0 lie inside the open interval (-2,1), so check those two.

   Point	Left limit	Value f(a)	Continuous?
   x=-1	[x]=-2, x-[x]→ 1: 2+√1=3	|-1|+√0=1	No, 3 ≠ 1
   x=0	[x]=-1, x-[x]→ 1: 1+√1=2	|0|+√0=0	No, 2 ≠ 0

2. Why it breaks. On the negative side, |[x]| drops by 1 as you cross an integer, and √(x-[x]) also drops from 1 to 0. The two drops add instead of cancelling, so there is a jump of 2 at each point.
3. So. f is discontinuous at x=-1 and x=0, so the number of points is 2.

The idea to keep. With [x] in a function, only the integers can cause trouble. At each one, compare the left limit with the value; the right limit always equals the value.

Answer from the NTA final answer key. Solution by PrepWiser.