The domain of continuity of the function f(x) = tan x is

Options

A. `R-{npi, n inZ}`
B. `R-{2npi, n inZ}`
C. `R-{(2n+1)(pi)/(2), n in Z}`
D. `R-{(npi)/(2),n in Z}`

Answer

To determine the domain of continuity of the function \( f(x) = \tan x \), we need to analyze where this function is defined and continuous. Here’s a step-by-step solution:

### Step 1: Understand the function
The function \( f(x) = \tan x \) can be expressed in terms of sine and cosine:
\[ f(x) = \tan x = \frac{\sin x}{\cos x} \]

### Step 2: Identify where the function is undefined
The function \( \tan x \) is undefined wherever the denominator \( \cos x \) is equal to zero. Therefore, we need to find the values of \( x \) for which:
\[ \cos x = 0 \]

### Step 3: Solve for \( x \)
The cosine function is equal to zero at the points:
\[ x = \frac{\pi}{2} + n\pi \quad \text{for } n \in \mathbb{Z} \]
This can be rewritten as:
\[ x = \frac{(2n + 1)\pi}{2} \quad \text{for } n \in \mathbb{Z} \]
These points are where \( \tan x \) is undefined.

### Step 4: Determine the domain of continuity
Since \( \tan x \) is continuous wherever it is defined, the function is continuous for all real numbers except at the points where \( \cos x = 0 \). Thus, the domain of continuity of \( f(x) = \tan x \) is:
\[ \mathbb{R} \setminus \left\{ \frac{(2n + 1)\pi}{2} \mid n \in \mathbb{Z} \right\} \]

### Step 5: Conclusion
The domain of continuity of the function \( f(x) = \tan x \) can be expressed in interval notation as:
\[ \text{Domain of continuity} = \mathbb{R} \setminus \left\{ \frac{(2n + 1)\pi}{2} \mid n \in \mathbb{Z} \right\} \]