Statement-1: The function `f(x)=[x]+x^(2)` is discontinuous at all integer points.
Statement-2: The function g(x)=[x] has Z as the set of points of its discontinuous from left.

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To analyze the statements given in the question, we will evaluate each statement step by step.

### Step 1: Evaluate Statement-2
The function \( g(x) = [x] \) is the greatest integer function, which returns the largest integer less than or equal to \( x \).

**Discontinuity of \( g(x) \):**
- The function \( g(x) \) is discontinuous at every integer point. This is because as \( x \) approaches an integer \( n \) from the left (i.e., \( n - 0.1, n - 0.01, \ldots \)), \( g(x) \) will return \( n - 1 \).
- However, at \( x = n \), \( g(n) = n \). Thus, there is a jump in the function value at every integer point.

**Conclusion for Statement-2:**
- Therefore, Statement-2 is correct: \( g(x) \) has \( \mathbb{Z} \) (the set of integers) as the set of points of its discontinuity from the left.

### Step 2: Evaluate Statement-1
The function \( f(x) = [x] + x^2 \) consists of two parts: the greatest integer function \( [x] \) and the continuous function \( x^2 \).

**Discontinuity of \( f(x) \):**
- Since \( [x] \) is discontinuous at every integer point, we need to check if the sum \( f(x) = [x] + x^2 \) is also discontinuous at those points.
- At an integer point \( n \):
- As \( x \) approaches \( n \) from the left, \( f(n - \epsilon) = [n - \epsilon] + (n - \epsilon)^2 = n - 1 + (n - \epsilon)^2 \).
- As \( x \) approaches \( n \) from the right, \( f(n + \epsilon) = [n + \epsilon] + (n + \epsilon)^2 = n + (n + \epsilon)^2 \).

**Limit Analysis:**
- The left-hand limit as \( x \to n \) is:
\[ \lim_{x \to n^-} f(x) = n - 1 + n^2 - 2n\epsilon + \epsilon^2 \quad \text{(approaches } n^2 - n + 1\text{)} \]
- The right-hand limit as \( x \to n \) is:
\[ \lim_{x \to n^+} f(x) = n + n^2 + 2n\epsilon + \epsilon^2 \quad \text{(approaches } n^2 + n\text{)} \]

Since the left-hand limit and right-hand limit at \( n \) are not equal, \( f(x) \) is discontinuous at every integer point.

**Conclusion for Statement-1:**
- Therefore, Statement-1 is also correct: the function \( f(x) = [x] + x^2 \) is discontinuous at all integer points.

### Final Conclusion
Both statements are correct.

### Summary of Steps:
1. Analyze the discontinuity of \( g(x) = [x] \) at integer points.
2. Evaluate the function \( f(x) = [x] + x^2 \) and check its continuity at integer points.
3. Conclude that both statements are correct.