The function f(x) = cot x is discountinuous on these

Options

A. `{x=npi, n inZ}`
B. `{x=2npi, n inZ}`
C. `{x=(xpi)/(2), n inZ}`
D. `{x=(2n+1)(xpi)/(2), n inZ}`

Answer

To determine the points of discontinuity for the function \( f(x) = \cot x \), we will follow these steps:

### Step 1: Understand the function
The cotangent function is defined as:
\[ f(x) = \cot x = \frac{\cos x}{\sin x} \]
The function is defined wherever the denominator is not zero.

### Step 2: Identify where the function is undefined
The function \( \cot x \) will be discontinuous wherever \( \sin x = 0 \) because division by zero is undefined.

### Step 3: Solve for points where \( \sin x = 0 \)
The sine function is equal to zero at integer multiples of \( \pi \):
\[ \sin x = 0 \implies x = n\pi \quad \text{where } n \text{ is any integer.} \]

### Step 4: Conclusion
Thus, the function \( f(x) = \cot x \) is discontinuous at:
\[ x = n\pi \quad \text{for } n \in \mathbb{Z} \]

### Final Answer
The function \( f(x) = \cot x \) is discontinuous at \( x = n\pi \) where \( n \) is an integer.

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