Question

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

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

Hint:

To find discontinuities of a function involving the floor function, check points where \( [x] \) changes, typically at integers. Compare left and right limits to confirm jumps.

Answer 1

The Correct Option is D

Step 1: Function Definition and Understanding.
The function is \( f(x) = x - [x] \), where \( [x] \) represents the floor function (greatest integer less than or equal to \( x \)). The fractional part of \( x \) is defined as \( \{x\} = x - [x] \), therefore: \[ f(x) = \{x\}, \] which is the fractional part of \( x \), with \( 0 \leq \{x\}<1 \) for all \( x \in \mathbb{R} \).
Step 2: Identifying Discontinuities.
A function is discontinuous where the limit doesn't match the function's value or the limit doesn't exist. For \( f(x) = \{x\} \):
At integer points \( x = n \) (where \( n \in \mathbb{Z} \)), \( [n] = n \), hence \( f(n) = n - n = 0 \).
As \( x \) approaches \( n \) from the left (\( x \to n^- \)), \( [x] = n - 1 \), so \( f(x) = x - (n - 1) \to n - (n - 1) = 1 \).
As \( x \) approaches \( n \) from the right (\( x \to n^+ \)), \( [x] = n \), giving \( f(x) = x - n \to 0 \).
Since the left-hand limit is 1, the right-hand limit is 0, and \( f(n) = 0 \), the limit doesn't exist at \( x = n \), indicating a discontinuity.
Step 3: Checking Non-Integer Points.
For \( x \) that aren't integers (e.g., \( x = n + \delta \) where \( 0<\delta<1 \)):
\( [x] = n \), so \( f(x) = x - n \).
As \( x \to (n + \delta)^- \) and \( x \to (n + \delta)^+ \), \( [x] \) remains \( n \) (because \( \delta \) is fixed and small), and \( f(x) = x - n \) is continuous because \( [x] \) is constant in a small neighborhood excluding integers.
Thus, \( f(x) \) is continuous at non-integer points.
Step 4: Determining the Discontinuity Set.
Discontinuities occur exclusively at integer points \( x = n \) where \( n \in \mathbb{Z} \). Therefore, the set of discontinuities is \( \mathbb{Z} \).
Step 5: Verification of Options.
% Option (A) \( \mathbb{Q} \): Rational numbers include non-integers where \( f(x) \) is continuous.
% Option (B) \( \mathbb{R} \): All real numbers include non-integers where \( f(x) \) is continuous.
% Option (C) \( \mathbb{N} \): Natural numbers are a subset of \( \mathbb{Z} \), but \( \mathbb{Z} \) encompasses all integers.
% Option (D) \( \mathbb{Z} \): Integers precisely match the points of discontinuity.