Prove that the Greatest Integer Function `f : R->R ,`given by `f (x) = [x]`, is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Explanation

To prove that the Greatest Integer Function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = [x] \) (where \([x]\) denotes the greatest integer less than or equal to \( x \)) is neither one-one nor onto, we will analyze both properties step by step.

### Step 1: Proving that \( f \) is not one-one (injective)

A function \( f \) is said to be one-one if \( f(a) = f(b) \) implies \( a = b \) for all \( a, b \in \mathbb{R} \).

**Example:**
Consider the values:
...