At how many points is the function `f(x)=[x]` discontinuous?

Options

A. 1
B. 2
C. 3
D. `oo`

Answer

To determine the points of discontinuity of the function \( f(x) = [x] \), where \([x]\) denotes the greatest integer less than or equal to \( x \), we will analyze the behavior of the function at various points.

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(x) = [x] \) gives the greatest integer less than or equal to \( x \). For example:
- If \( x = 2.3 \), then \( f(2.3) = 2 \).
- If \( x = -1.7 \), then \( f(-1.7) = -2 \).

2. **Identifying the Discontinuities**:
The function \( f(x) \) is piecewise constant between integers. It changes value at every integer point. We need to check the behavior of the function at these integer points.

3. **Evaluating Limits at Integer Points**:
- At \( x = 0 \):
- Left-hand limit (LHL): \( \lim_{x \to 0^-} f(x) = -1 \)
- Right-hand limit (RHL): \( \lim_{x \to 0^+} f(x) = 0 \)
- Since LHL ≠ RHL, \( f(x) \) is discontinuous at \( x = 0 \).

- At \( x = 1 \):
- LHL: \( \lim_{x \to 1^-} f(x) = 0 \)
- RHL: \( \lim_{x \to 1^+} f(x) = 1 \)
- Since LHL ≠ RHL, \( f(x) \) is discontinuous at \( x = 1 \).

- Similarly, we can check for other integer points:
- At \( x = -1 \):
- LHL: \( \lim_{x \to -1^-} f(x) = -2 \)
- RHL: \( \lim_{x \to -1^+} f(x) = -1 \)
- Discontinuous at \( x = -1 \).

4. **Generalizing the Result**:
This pattern continues for all integer values. At every integer \( n \), the function \( f(x) \) is discontinuous because the left-hand limit and right-hand limit do not match.

5. **Conclusion**:
Since there are infinitely many integers, the function \( f(x) = [x] \) is discontinuous at every integer point.

### Final Answer:
The function \( f(x) = [x] \) is discontinuous at an infinite number of points.

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