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 follow these steps:

### Step 1: Prove that the function is not one-one (injective)

**Explanation:**
A function is one-one (or injective) if different inputs map to different outputs. In other words, if \( f(a) = f(b) \) implies \( a = b \).

**Example:**
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