Prove that f(x)=x-[x] is periodic function. Also, find its period.

Answer

To prove that the function \( f(x) = x - [x] \) is periodic and to find its period, we will follow these steps:

### Step 1: Understanding the function
The function \( f(x) = x - [x] \) represents the fractional part of \( x \), where \( [x] \) denotes the greatest integer less than or equal to \( x \). Thus, \( f(x) \) gives us the decimal part of \( x \).

### Step 2: Analyzing the function
We can rewrite the function as:
\[ f(x) = x - [x] = \{ x \} \]
where \( \{ x \} \) is the fractional part of \( x \). This function takes any real number \( x \) and subtracts the integer part, leaving only the fractional part.

### Step 3: Checking periodicity
To check if \( f(x) \) is periodic, we need to find a value \( T \) such that:
\[ f(x + T) = f(x) \quad \text{for all } x \]
Let's consider \( T = 1 \):
\[ f(x + 1) = (x + 1) - [x + 1] \]
Using the property of the greatest integer function:
\[ [x + 1] = [x] + 1 \]
Thus:
\[ f(x + 1) = (x + 1) - ([x] + 1) = x - [x] = f(x) \]
This shows that:
\[ f(x + 1) = f(x) \]
for all \( x \).

### Step 4: Conclusion about periodicity
Since \( f(x + 1) = f(x) \) holds for all \( x \), we conclude that \( f(x) \) is periodic with period \( T = 1 \).

### Step 5: Final answer
Therefore, we can state that:
- \( f(x) = x - [x] \) is a periodic function.
- The period of the function is \( 1 \).