Question

The set of points of discontinuity of the function \[ f(x) = x - [x], \quad x \in \mathbb{R} \] is:

• A. \( \mathbb{Q} \)
• B. \( \mathbb{R} \)
• C. \( \mathbb{N} \)
• D. \( \mathbb{Z} \)

Hint:

For fractional part function: \( \{x\} = x - [x] \) Continuous on intervals \( (n, n+1) \) Jump discontinuity at every integer.

Answer 1

The Correct Option is D

Concept: The function: \[ f(x) = x - [x] \] is the fractional part function, denoted by: \[ \{x\} \] Properties:

• 0 ≤ {x} < 1
• Continuous everywhere except at integers

Step 1: Understand floor function behavior. The greatest integer function \( [x] \) has jump discontinuities at integers. Hence: \[ x - [x] \] also becomes discontinuous at integers. Step 2: Check continuity elsewhere. Between integers, the floor value remains constant, so the function behaves like: \[ f(x) = x - \text{constant} \] which is continuous. Step 3: Final conclusion. Discontinuities occur at: \[ \mathbb{Z} \]