Let R be the set of all real numbers and f : R → R be a continuous function. Suppose | f ( x ) - f ( y ) | ≥ | x - y | for all real numbers x and y . Then

Step-by-step solution

Let f ( x ) = f ( y ) So, | f ( x ) - f ( y ) | ≥ | x - y | ⇒ 0 ≥ | x - y | ⇒ x - y = 0 ⇒ x = y ⇒ f is one-one Since, f is continuous So f ( 0 ) is finite Now, | f ( x ) - f ( 0 ) | ≥ | x - 0 | ⇒ lim x → ∞ | f ( x ) - f ( 0 ) | ≥ lim x → ∞ | x | ⇒ lim x → ∞ f ( x ) = ∞ ⇒ f is unbounded ⇒ f is surjective