Question

For \( f(x) = [x] \), where \( [x] \) is the greatest integer function, which of the following is true, for every \( x \in \mathbb{R} \)?

• A. \( [x] + 1 = x \)
• B. \( [x] + 1 \leq x \)
• C. \( [x] + 1>x \)
• D. \( [x] + 1<x \)

Hint:

For the greatest integer function \( [x] \), the relation \( [x] \leq x \) always holds, and adding 1 to \( [x] \) makes it strictly less than \( x \) when \( x \) is not an integer.

Answer 1

The Correct Option is D

Step 1: Understanding the greatest integer function.
The greatest integer function \( [x] \) returns the greatest integer less than or equal to \( x \). For example, for \( x = 2.5 \), \( [x] = 2 \), and for \( x = -1.5 \), \( [x] = -2 \). Therefore, \( [x] \leq x \).

Step 2: Analyzing the options.
- \( [x] + 1<x \) is the correct relationship because \( [x] \) is always less than or equal to \( x \), and adding 1 to \( [x] \) makes it strictly less than \( x \) for non-integer values.

Step 3: Conclusion.
Thus, the correct answer is \( [x] + 1<x \), corresponding to option (D).