Let [ ] denote the greatest integer function and `f(x) = [tan^(2)x]` Then

Options

A. `lim_(x to 0) f(x)` does not exist
B. f(x) is continuous at x = 0
C. f(x) is not differentiable at x = 0
D. `f'(0)=1`

Answer

To solve the problem, we need to analyze the function \( f(x) = \lfloor \tan^2 x \rfloor \) where \( \lfloor \cdot \rfloor \) denotes the greatest integer function. We will focus on the interval \( x = -45^\circ \) to \( x = 45^\circ \).

### Step-by-Step Solution:

1. **Identify the Range of \( \tan^2 x \)**:
- The tangent function \( \tan x \) is defined and continuous in the interval \( (-45^\circ, 45^\circ) \).
- At the endpoints, we have:
- \( \tan(-45^\circ) = -1 \)
- \( \tan(45^\circ) = 1 \)
- Therefore, \( -1 < \tan x < 1 \) for \( -45^\circ < x < 45^\circ \).

2. **Square the Tangent Function**:
- Squaring the tangent function gives:
\[ 0 < \tan^2 x < 1 \]
- This means that \( \tan^2 x \) takes values between 0 and 1.

3. **Apply the Greatest Integer Function**:
- Since \( 0 < \tan^2 x < 1 \), applying the greatest integer function gives:
\[ f(x) = \lfloor \tan^2 x \rfloor = 0 \]
- This holds true for all \( x \) in the interval \( (-45^\circ, 45^\circ) \).

4. **Check Continuity**:
- The function \( f(x) = 0 \) is a constant function in the interval \( (-45^\circ, 45^\circ) \).
- A constant function is continuous everywhere, hence \( f(x) \) is continuous in this interval.

5. **Check Differentiability**:
- The derivative of a constant function is zero. Therefore, \( f'(x) = 0 \) for all \( x \) in the interval.
- However, at points where \( \tan^2 x \) approaches 1 (specifically at \( x = 45^\circ \) or \( x = -45^\circ \)), the function is not defined due to the discontinuity of \( \tan x \) at these points.

6. **Conclusion**:
- The function \( f(x) \) is continuous and differentiable for all \( x \) in the interval \( (-45^\circ, 45^\circ) \) except at the endpoints where \( \tan x \) is undefined.

### Final Answer:
- The function \( f(x) = \lfloor \tan^2 x \rfloor \) is continuous and differentiable in the interval \( (-45^\circ, 45^\circ) \) except at the endpoints.