The function `f(x)=(tan |pi[x-pi]|)/(1+[x]^(2))`, where [x] denotes the greatest integer less than or equal to x, is

Options

A. f(x) is discontinuous at some x
B. f(x) is continuous at all x, but the derivative f'(x) does not exist for some x
C. `f'(x)` exists for all x, but f''(x) does not exist for some x
D. f'(x) exists for all x

Answer

To analyze the function \( f(x) = \frac{\tan(\pi |x - \pi|)}{1 + [x]^2} \), where \([x]\) denotes the greatest integer less than or equal to \(x\), we will go through the following steps:

### Step 1: Understanding the Function
We start with the function:
\[ f(x) = \frac{\tan(\pi |x - \pi|)}{1 + [x]^2} \]
Here, \(|x - \pi|\) is the absolute value of \(x - \pi\) and \([x]\) is the greatest integer function.

### Step 2: Analyzing the Numerator
The numerator is \(\tan(\pi |x - \pi|)\). The tangent function is periodic with period \(\pi\):
- For any integer \(n\), \(\tan(n\pi) = 0\).
- Therefore, \(\tan(\pi |x - \pi|) = 0\) whenever \(|x - \pi| = n\) for any integer \(n\).

### Step 3: Analyzing the Denominator
The denominator is \(1 + [x]^2\):
- The greatest integer function \([x]\) is always an integer, so \([x]^2\) is non-negative.
- Thus, \(1 + [x]^2 \geq 1\) for all \(x\), meaning it is always positive and never zero.

### Step 4: Simplifying the Function
Since the numerator \(\tan(\pi |x - \pi|) = 0\) for all \(x\) (as \(|x - \pi|\) will take values that make the tangent zero), we can conclude:
\[ f(x) = \frac{0}{1 + [x]^2} = 0 \]
for all \(x\).

### Step 5: Conclusion about Continuity and Differentiability
Since \(f(x) = 0\) for all \(x\):
- \(f(x)\) is a constant function.
- A constant function is continuous everywhere.
- The derivative \(f'(x) = 0\) exists for all \(x\).
- The second derivative \(f''(x) = 0\) also exists for all \(x\).

### Final Answer
Thus, we conclude:
- The function \(f(x)\) is continuous and differentiable everywhere, and both the first and second derivatives exist for all \(x\).

### Summary of Options
1. **Incorrect**: \(f'(x)\) does exist for all \(x\).
2. **Incorrect**: \(f(x)\) is continuous and \(f'(x)\) exists.
3. **Incorrect**: \(f''(x)\) exists for all \(x\).
4. **Correct**: \(f(x)\) is a constant function.