Let `[x]` be the greatest integer less than or equal to `xdot` Then, at which of the following point (s) function `f(x)=xcos(pi(x+[x]))` is discontinuous? (a)`x=1` (b) `x=-1` (c) `x=0` (d) `x=2`

Explanation

To determine the points of discontinuity for the function \( f(x) = x \cos(\pi (x + [x])) \), where \([x]\) is the greatest integer less than or equal to \(x\), we will analyze the function at the given points: \(x = 1\), \(x = -1\), \(x = 0\), and \(x = 2\).

### Step-by-Step Solution:

1. **Check for discontinuity at \(x = 1\)**:
- **Left-hand limit**:
\[
\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x \cos(\pi (x + [x])) = \lim_{x \to 1^-} x \cos(\pi (x + 0)) = 1 \cdot \cos(\pi \cdot 1) = 1 \cdot (-1) = -1
...