Let `[.]` represent the greatest integer function and `f (x)=[tan^2 x]` then :

Options

A. `lim _(xto0) 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)=0`

Answer

To solve the problem, we need to analyze the function \( f(x) = [\tan^2 x] \), where \([.]\) denotes the greatest integer function. We will evaluate the limit of \( f(x) \) as \( x \) approaches 0 from both the left and the right.

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(x) = [\tan^2 x] \) means we take the square of the tangent of \( x \) and then apply the greatest integer function to it.

2. **Finding the Left-Hand Limit**:
We will first find the limit as \( x \) approaches 0 from the left (\( x \to 0^- \)):
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} [\tan^2 x] \]
As \( x \) approaches 0 from the left, \( \tan x \) approaches 0. Therefore, \( \tan^2 x \) also approaches 0:
\[ \lim_{x \to 0^-} \tan^2 x = 0 \implies \lim_{x \to 0^-} f(x) = [0] = 0 \]

3. **Finding the Right-Hand Limit**:
Now we find the limit as \( x \) approaches 0 from the right (\( x \to 0^+ \)):
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} [\tan^2 x] \]
Similarly, as \( x \) approaches 0 from the right, \( \tan x \) approaches 0, hence:
\[ \lim_{x \to 0^+} \tan^2 x = 0 \implies \lim_{x \to 0^+} f(x) = [0] = 0 \]

4. **Checking Continuity**:
Since both the left-hand limit and right-hand limit as \( x \) approaches 0 are equal:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = 0 \]
We also need to check the value of \( f(0) \):
\[ f(0) = [\tan^2(0)] = [0] = 0 \]
Therefore, since \( \lim_{x \to 0} f(x) = f(0) \), the function \( f(x) \) is continuous at \( x = 0 \).

5. **Differentiability**:
Next, we need to check if \( f(x) \) is differentiable at \( x = 0 \). The greatest integer function is not differentiable at integer points. Since \( f(x) \) is a greatest integer function, it is not differentiable at \( x = 0 \).

6. **Conclusion**:
- \( f(x) \) is continuous at \( x = 0 \).
- \( f(x) \) is not differentiable at \( x = 0 \).

### Final Answer:
The correct option is that \( f(x) \) is continuous at \( x = 0 \) but not differentiable at \( x = 0 \).