Plot the following, where [.] denotes integer function.
`f(x)=cos(x-[x])`

Answer

To plot the function \( f(x) = \cos(x - [x]) \), where \([x]\) denotes the greatest integer function (also known as the floor function), we can follow these steps:

### Step 1: Understand the function
The expression \( x - [x] \) represents the fractional part of \( x \). Therefore, we can rewrite the function as:
\[ f(x) = \cos(x - [x]) = \cos(\{x\}) \]
where \(\{x\} = x - [x]\) is the fractional part of \( x \).

### Step 2: Determine the range of the fractional part
The fractional part \(\{x\}\) takes values in the interval \([0, 1)\). This means that as \( x \) varies, \(\{x\}\) will cycle through values from 0 to just below 1.

### Step 3: Analyze the cosine function
The cosine function, \( \cos(t) \), has a range of \([-1, 1]\) and is periodic with a period of \( 2\pi \). However, since we are only interested in the interval \([0, 1)\) for \(\{x\}\), we will evaluate \( \cos(t) \) for \( t \) in this range.

### Step 4: Calculate specific values
1. At \( x = 0 \):
\[ f(0) = \cos(0) = 1 \]
2. As \( x \) approaches 1 (but does not reach it):
\[ f(1) = \cos(0) = 1 \quad \text{(open interval at 1)} \]
3. At \( x = 0.5 \):
\[ f(0.5) = \cos(0.5) \approx 0.8776 \]
4. At \( x = 0.9 \):
\[ f(0.9) = \cos(0.9) \approx 0.6216 \]

### Step 5: Plot the function for intervals
The function will repeat its behavior for each integer interval. For each integer \( n \):
- From \( n \) to \( n+1 \), \( f(x) \) will take values from \( \cos(0) \) to \( \cos(1) \) and will return to \( \cos(0) \) at \( n+1 \) (open interval).

### Step 6: Sketch the graph
1. For \( x \) in each interval \([n, n+1)\), the graph will start at \( f(n) = 1 \), decrease to \( f(n+1) = 1 \) (open).
2. The same behavior will occur for negative integers, mirroring the positive side.

### Final Graph
The graph will consist of segments that start at \( (n, 1) \), curve downwards, and approach \( (n+1, 1) \) without touching it, repeating this pattern for all integers \( n \).