Which of the following function are not periodic (where [.] denotes greatest integer function:
`f(x)=x+sinx`

Explanation

To determine whether the function \( f(x) = x + \sin x \) is periodic, we will follow these steps:

### Step 1: Understand the definition of a periodic function
A function \( f(x) \) is said to be periodic if there exists a positive number \( T \) (the period) such that:
\[ f(x + T) = f(x) \quad \text{for all } x \]

### Step 2: Substitute \( x + T \) into the function
We will evaluate \( f(x + T) \):
\[ f(x + T) = (x + T) + \sin(x + T) \]
This simplifies to:
\[ f(x + T) = x + T + \sin(x + T) \]

### Step 3: Compare \( f(x + T) \) with \( f(x) \)
Now we compare \( f(x + T) \) with \( f(x) \):
\[ f(x) = x + \sin x \]
For \( f(x + T) \) to equal \( f(x) \), we need:
\[ x + T + \sin(x + T) = x + \sin x \]
This simplifies to:
\[ T + \sin(x + T) = \sin x \]

### Step 4: Analyze the equation
The equation \( T + \sin(x + T) = \sin x \) implies that:
\[ \sin(x + T) = \sin x - T \]
The sine function is periodic with a period of \( 2\pi \), but the term \( T \) is a constant. As \( x \) varies, \( \sin x \) oscillates between -1 and 1, while \( T \) is a fixed value. This means that \( \sin(x + T) \) will not be able to equal \( \sin x - T \) for all \( x \) unless \( T = 0 \).

### Step 5: Conclusion
Since we cannot find a positive \( T \) such that \( f(x + T) = f(x) \) for all \( x \), the function \( f(x) = x + \sin x \) is not periodic.

### Final Answer
Thus, the function \( f(x) = x + \sin x \) is **not periodic**.

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