The function `f(x)=xcosecx` is

Options

A. continuous at all value of x
B. discontinuous everywhere
C. continuous for all x except at `x=npi` where n is an integer
D. continuous for all x except at `x=npi//2` where n is an integer

Answer

To determine the continuity of the function \( f(x) = x \cos x \), we need to analyze the function carefully.

### Step-by-Step Solution:

1. **Identify the Function**:
The function given is \( f(x) = x \cos x \).

2. **Understand Continuity**:
A function is continuous at a point if the limit of the function as it approaches that point is equal to the function's value at that point. For a function to be continuous everywhere, it must be continuous at every point in its domain.

3. **Check for Discontinuities**:
The function \( f(x) = x \cos x \) is a product of \( x \) (a polynomial function) and \( \cos x \) (a trigonometric function). Both of these functions are continuous everywhere in their respective domains.

4. **Analyze the Denominator**:
The function can also be expressed in terms of sine:
\[ f(x) = \frac{x}{\sin x} \]
We need to check where \( \sin x = 0 \) because division by zero would cause discontinuity. The sine function is zero at integer multiples of \( \pi \):
\[ x = n\pi \quad (n \in \mathbb{Z}) \]

5. **Determine Points of Discontinuity**:
At points \( x = n\pi \), \( f(x) \) becomes undefined because the denominator \( \sin x \) is zero. Therefore, the function is discontinuous at these points.

6. **Conclusion**:
The function \( f(x) = x \cos x \) is continuous everywhere except at \( x = n\pi \) where \( n \) is an integer. Thus, the correct answer is that the function is continuous for all \( x \) except at \( x = n\pi \).

### Final Answer:
The function \( f(x) = x \cos x \) is continuous for all \( x \) except at \( x = n\pi \) where \( n \) is an integer.

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