Understanding the Fractional Part Function

The function given is \( f(x) = x - [x] \), where \( x \ge 0 \). The term \( [x] \) denotes the greatest integer function, which returns the largest integer less than or equal to \( x \). The function \( f(x) \) specifically calculates the fractional part of \( x \).

Analyzing Behavior Between Integers

Consider the function's behavior over an interval between two consecutive integers, say \( [n, n+1) \), where \( n \) is any non-negative integer.

• Within the interval \( [n, n+1) \), the value of the greatest integer function is constant: \( [x] = n \).
• Substituting this into the function definition, we get \( f(x) = x - n \) for \( x \in [n, n+1) \).
• This expression \( f(x) = x - n \) is a linear equation with a slope of \( 1 \).

Interpreting the Function's Nature

Because \( f(x) = x - n \) has a positive slope (\( 1 \)) within the interval \( [n, n+1) \), the function increases linearly as \( x \) increases in this range.

• At \( x = n \), \( f(x) = n - n = 0 \).
• As \( x \) increases towards \( n+1 \), \( f(x) \) increases. For example, if \( x \) is just below \( n+1 \), \( f(x) \) will be just below \( (n+1) - n = 1 \).

While the function drops to 0 at every integer value of \( x \) (since \( f(n+1) = (n+1) - [n+1] = (n+1) - (n+1) = 0 \)), its behavior *between* any two consecutive integers is consistently linear and increasing.

Therefore, the function \( f(x) = x - [x] \) is a linearly increasing function between two integers.