Let `f(x) = (sin (pi [ x + pi]))/(1+[x]^(2))` where [] denotes the greatest integer function then f(x) is

Options

A. continuous and differentiable at all ` x in R `
B. continuous but not differentiable at some x
C. Differentiable but not continuous at x =0
D. Neither continuous nor differentiable at x =0

Answer

To analyze the function \( f(x) = \frac{\sin(\pi [x + \pi])}{1 + [x]^2} \), where \([x]\) denotes the greatest integer function, we will determine the continuity and differentiability of \( f(x) \).

### Step 1: Understand the Greatest Integer Function
The greatest integer function \([x]\) returns the largest integer less than or equal to \( x \). For example:
- If \( x = 2.3 \), then \([x] = 2\).
- If \( x = 3.7 \), then \([x] = 3\).
- If \( x = 4 \), then \([x] = 4\).

### Step 2: Analyze the Function
The function can be rewritten as:
\[ f(x) = \frac{\sin(\pi ([x] + \pi))}{1 + [x]^2} \]
This means that the value of \( f(x) \) depends on the integer part of \( x \).

### Step 3: Identify Points of Discontinuity
The sine function, \(\sin(\pi n)\), where \( n \) is an integer, equals zero. Therefore, \( f(x) \) will be zero whenever \([x] + \pi\) is an integer, which occurs when \([x]\) is an integer. This means \( f(x) \) will be zero at integer values of \( x \).

### Step 4: Check Continuity
To check for continuity at integer points:
- As \( x \) approaches an integer \( n \) from the left (e.g., \( n - 0.1 \)), \([x] = n - 1\), and thus \( f(x) \) approaches:
\[ f(n - 0.1) = \frac{\sin(\pi (n - 1 + \pi))}{1 + (n - 1)^2} \]
- As \( x \) approaches \( n \) from the right (e.g., \( n + 0.1 \)), \([x] = n\), and thus \( f(x) \) approaches:
\[ f(n + 0.1) = \frac{\sin(\pi (n + \pi))}{1 + n^2} \]
- Both limits will yield different values, confirming that \( f(x) \) is discontinuous at integer points.

### Step 5: Check Differentiability
Since \( f(x) \) is discontinuous at integer points, it cannot be differentiable at those points. However, for all other points where \( x \) is not an integer, \( f(x) \) is continuous and differentiable because both the sine function and the polynomial in the denominator are continuous and differentiable.

### Conclusion
The function \( f(x) \) is:
- Discontinuous at integer values of \( x \).
- Continuous and differentiable for all \( x \) that are not integers.