Cosider the following statements: 1. f(x)=[x], where [.] is the greatest integer function, is discontinuous at x=n, where `ninZ`.
2. `f(x)=cotx` is discontinuous at `x=npi,` where `ninZ`.
Which of the above statements is /are correct?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To determine the correctness of the given statements, we need to analyze each statement one by one.

### Statement 1:
**Function:** \( f(x) = [x] \) (greatest integer function)

**Analysis:**
The greatest integer function, denoted as \([x]\), gives the largest integer less than or equal to \(x\). This function is known to be discontinuous at integer values. Specifically, at any integer \(n\), the function jumps from \([n-1]\) to \([n]\) as \(x\) approaches \(n\) from the left and right.

- For \(x\) approaching \(n\) from the left (\(x \to n^{-}\)), \([x] = n-1\).
- For \(x\) approaching \(n\) from the right (\(x \to n^{+}\)), \([x] = n\).

Since the left-hand limit does not equal the right-hand limit at \(x = n\), the function is discontinuous at \(x = n\).

**Conclusion for Statement 1:** The statement is correct.

### Statement 2:
**Function:** \( f(x) = \cot(x) \)

**Analysis:**
The cotangent function, \(\cot(x)\), is defined as \(\frac{\cos(x)}{\sin(x)}\). It is discontinuous wherever \(\sin(x) = 0\), which occurs at \(x = n\pi\) for any integer \(n\).

- At \(x = n\pi\), \(\sin(n\pi) = 0\), leading to \(\cot(n\pi)\) being undefined (or tending towards \(\pm \infty\)).

Thus, the function \(\cot(x)\) is indeed discontinuous at \(x = n\pi\).

**Conclusion for Statement 2:** The statement is correct.

### Final Conclusion:
Both statements are correct.

### Summary:
1. The first statement about the greatest integer function being discontinuous at integer points is correct.
2. The second statement about the cotangent function being discontinuous at integer multiples of \(\pi\) is also correct.

### Final Answer:
Both statements are correct.

---