JEE Mathematics 2023 Question with Solution

Given: $f(x) = |[x]| + \sqrt{x - [x]}$ on the interval $(-2,1)$.

Find: The number of points where the function is discontinuous in this interval.

The doubtful points are the integer points inside or at the boundary transition relevant to the interval, so we check $x=-1,0,1$ as shown in the extracted solution.

At $x=-1$:

So the left-hand and right-hand limits are unequal, hence $f(x)$ is discontinuous at $x=-1$.

At $x=0$:

Again the one-sided limits are unequal, so $f(x)$ is discontinuous at $x=0$.

At $x=1$:

So there is no discontinuity at $x=1$.

Therefore, discontinuity occurs at two points. Hence the answer is $2$.

The answer key states $3$, but the solution working concludes the correct count is $2$, so the solution is taken as authoritative.

Given: $f(x) = |[x]| + \sqrt{x-[x]}$.

Find: How many discontinuities lie in $(-2,1)$.

The greatest integer function changes value at integers, so discontinuities can occur only at integer points. In the relevant range, the extracted solution checks the points near the interval as follows:

1. At $x=-1$, the values from the two sides are different.
2. At $x=0$, the values from the two sides are different.
3. At $x=1$, the values from the two sides are equal.

Thus only $x=-1$ and $x=0$ are discontinuity points.

So the number of discontinuities is $2$.