The function `f(x)=min{x-[x],-x-[-x]}` is a

Options

A. periodic function with period 1
B. periodic function with period 1/2
C. non-periodic function
D. periodic function with period 2

Explanation

To analyze the function \( f(x) = \min\{x - [x], -x - [-x]\} \), we will break it down step by step.

### Step 1: Understanding the Components
The function consists of two parts:
1. \( x - [x] \): This represents the fractional part of \( x \), denoted as \( \{x\} \).
2. \( -x - [-x] \): This can be rewritten using the properties of the greatest integer function.

**Hint:** Recall that \( [x] \) is the greatest integer less than or equal to \( x \), and \( [-x] \) is the greatest integer less than or equal to \( -x \).

### Step 2: Simplifying the Second Part
The term \( -x - [-x] \) can be simplified:
- For any real number \( x \), \( -x \) will have a greatest integer \( [-x] \).
- Thus, \( -x - [-x] = -x - (-[x] - 1) = -x + [x] + 1 \).

So, we can rewrite the function as:
\[ f(x) = \min\{\{x\}, -x + [x] + 1\} \]

**Hint:** Understand how the greatest integer function behaves with negative values.

### Step 3: Analyzing the Function
Now we analyze the two components:
1. The fractional part \( \{x\} \) is periodic with a period of 1 and ranges from 0 to 1.
2. The expression \( -x + [x] + 1 \) behaves differently depending on the integer part of \( x \).

**Hint:** Consider how \( \{x\} \) and \( -x + [x] + 1 \) behave in different intervals.

### Step 4: Finding Periodicity
To find the periodicity of \( f(x) \):
- Notice that \( f(x + 1) = \min\{(x + 1) - [x + 1], -(x + 1) - [- (x + 1)]\} \).
- Since both components \( \{x\} \) and \( -x + [x] + 1 \) are periodic with period 1, we conclude that \( f(x) \) is also periodic with period 1.

**Hint:** Check values of \( f(x) \) for \( x \) in the interval [0, 1] and see how they repeat for \( x + 1 \).

### Step 5: Conclusion
Thus, the function \( f(x) \) is periodic with a period of 1.

### Final Answer
The function \( f(x) = \min\{x - [x], -x - [-x]\} \) is periodic with a period of 1.

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