The function f(x)=(x) where (x) denotes the smallest integer `ge x` is

Options

A. everywhere continuous
B. continuous at x=n, `n in Z`
C. continuous on R-Z
D. none of these

Explanation

To analyze the function \( f(x) = \lceil x \rceil \), where \( \lceil x \rceil \) denotes the smallest integer greater than or equal to \( x \), we will determine the continuity of this function.

### Step 1: Understand the function
The function \( f(x) = \lceil x \rceil \) is defined as the smallest integer that is greater than or equal to \( x \). For example:
- If \( x = 2.3 \), then \( f(2.3) = 3 \)
- If \( x = -1.5 \), then \( f(-1.5) = -1 \)
- If \( x = 2 \), then \( f(2) = 2 \)

### Step 2: Graph the function
To visualize the function, we can sketch the graph of \( f(x) \):
- The function will have horizontal segments between each pair of integers.
- For any integer \( n \), \( f(x) \) will be equal to \( n \) for \( x \) in the interval \( [n, n+1) \).
- At each integer point \( n \), the function jumps to \( n \).

### Step 3: Identify points of discontinuity
The function \( f(x) = \lceil x \rceil \) is discontinuous at every integer point. This is because:
- As \( x \) approaches an integer \( n \) from the left (i.e., \( x \to n^- \)), \( f(x) \) approaches \( n \).
- At the integer \( n \), \( f(n) = n \).
- As \( x \) approaches \( n \) from the right (i.e., \( x \to n^+ \)), \( f(x) \) jumps to \( n+1 \).

Thus, there is a jump discontinuity at every integer \( n \).

### Step 4: Determine continuity on intervals
The function is continuous on the intervals between the integers:
- For any interval \( (n, n+1) \) where \( n \) is an integer, \( f(x) \) is constant and equal to \( n+1 \).
- Therefore, \( f(x) \) is continuous on \( \mathbb{R} \setminus \mathbb{Z} \) (the set of real numbers excluding integers).

### Conclusion
Based on the analysis, we conclude that:
- The function \( f(x) = \lceil x \rceil \) is **not continuous** at any integer point.
- The function is **continuous** on \( \mathbb{R} \setminus \mathbb{Z} \).

### Final Answer
The correct option is: **Continuous on \( \mathbb{R} \setminus \mathbb{Z} \)**.

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