Find the period of the function
`f(x) = tan""(pi)/(2) [x]`. Where [*] denotes greatest integer function

Answer

To find the period of the function \( f(x) = \tan\left(\frac{\pi}{2} [x]\right) \), where \([x]\) denotes the greatest integer function, we can follow these steps:

### Step 1: Understand the function
The function \( f(x) \) involves the tangent function and the greatest integer function. The greatest integer function, \([x]\), gives the largest integer less than or equal to \( x \).

### Step 2: Analyze the greatest integer function
The greatest integer function \([x]\) takes on integer values. For example:
- If \( 0 \leq x < 1 \), then \([x] = 0\)
- If \( 1 \leq x < 2 \), then \([x] = 1\)
- If \( 2 \leq x < 3 \), then \([x] = 2\)
- If \( 3 \leq x < 4 \), then \([x] = 3\)
- And so on...

### Step 3: Substitute values into the function
Now, we can substitute these values into the function:
- For \( 0 \leq x < 1 \):
\[ f(x) = \tan\left(\frac{\pi}{2} \cdot 0\right) = \tan(0) = 0 \]
- For \( 1 \leq x < 2 \):
\[ f(x) = \tan\left(\frac{\pi}{2} \cdot 1\right) = \tan\left(\frac{\pi}{2}\right) \text{ (undefined, or } \infty\text{)} \]
- For \( 2 \leq x < 3 \):
\[ f(x) = \tan\left(\frac{\pi}{2} \cdot 2\right) = \tan(\pi) = 0 \]
- For \( 3 \leq x < 4 \):
\[ f(x) = \tan\left(\frac{\pi}{2} \cdot 3\right) = \tan\left(\frac{3\pi}{2}\right) \text{ (undefined, or } \infty\text{)} \]

### Step 4: Identify the pattern
From the calculations:
- For intervals \( [0, 1) \) and \( [2, 3) \), the function outputs \( 0 \).
- For intervals \( [1, 2) \) and \( [3, 4) \), the function outputs \( \infty \).

### Step 5: Determine the period
The function repeats its values every 2 units:
- \( f(x) \) is \( 0 \) for \( [0, 1) \) and \( [2, 3) \)
- \( f(x) \) is \( \infty \) for \( [1, 2) \) and \( [3, 4) \)

Thus, the period of the function \( f(x) \) is \( 2 \).

### Final Answer
The period of the function \( f(x) = \tan\left(\frac{\pi}{2} [x]\right) \) is \( 2 \).

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