The function `f(x)=x-[x]` is a periodic with period.

Options

A. 1
B. 2
C. 3
D. none of these

Explanation

To determine the period of the function \( f(x) = x - [x] \), where \([x]\) denotes the greatest integer less than or equal to \(x\), we can follow these steps:

### Step 1: Understand the function
The function \( f(x) = x - [x] \) represents the fractional part of \(x\). This means it gives the non-integer part of \(x\).

### Step 2: Analyze the fractional part
The fractional part of any real number \(x\) is defined as:
\[ f(x) = x - [x] \]
This value will always lie in the interval \([0, 1)\). For example:
- If \(x = 2.3\), then \([x] = 2\) and \(f(2.3) = 2.3 - 2 = 0.3\).
- If \(x = 3.7\), then \([x] = 3\) and \(f(3.7) = 3.7 - 3 = 0.7\).

### Step 3: Determine periodicity
A function is periodic if there exists a positive number \(T\) such that:
\[ f(x + T) = f(x) \quad \text{for all } x \]
We will check if \(f(x + 1) = f(x)\):
\[ f(x + 1) = (x + 1) - [x + 1] = (x + 1) - ([x] + 1) = x - [x] = f(x) \]
This shows that \(f(x + 1) = f(x)\).

### Step 4: Conclusion
Since \(f(x + 1) = f(x)\) for all \(x\), we conclude that the function \(f(x)\) is periodic with a period of \(1\).

Thus, the answer is:
\[ \text{The period of the function } f(x) = x - [x] \text{ is } 1. \]

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