The number of points of discontinuity of the function $f(x)=\left[\frac{x^2}{2}\right]-[\sqrt{x}]$, $x \in [0,4]$, where $[\cdot]$ denotes the greatest integer function is

JEE Main · 2025 · 07 Apr 2025 · Shift 1 · Continuity and Differentiability

The number of points of discontinuity of the function

f (x) = [\frac{x ^{2}}{2}] - [x]

x \in [0, 4]

, where

[\cdot]

denotes the greatest integer function is

Answer: 8

The function changes value only where one of the floor terms jumps. On

[0, 4]

the term

[\frac{x ^{2}}{2}]

reaches a new integer at

x = 2, 2, 6, 8, 10, 12, 14

and at

x = 4

, while

[x]

jumps at

x = 1

and

x = 4

. At

x = 4

both terms rise by

1

, so their difference is unchanged and the function stays continuous there. The remaining points

1, 2, 2, 6, 8, 10, 12, 14

are genuine jumps, so the function is discontinuous at

8

points.

JEE Main · 2025 · 07 Apr 2025 · Shift 1 · Integer answer