If `f(x) = [x] , 0<= {x} < 0.5 and f(x) = [x]+1 , 0.5<{x}<1 ` then prove that f (x) = -f(-x) (where[.] and{.} represent the greatest integer function and the fractional part function, respectively).

Explanation

To prove that \( f(x) = -f(-x) \) for the given piecewise function, we will analyze the function \( f(x) \) based on the definition provided.

### Step 1: Define the function \( f(x) \)
The function \( f(x) \) is defined as:
- \( f(x) = [x] \) when \( 0 \leq \{x\} < 0.5 \)
- \( f(x) = [x] + 1 \) when \( 0.5 \leq \{x\} < 1 \)

Where \( [x] \) is the greatest integer function (floor function) and \( \{x\} \) is the fractional part of \( x \).
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