How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

Real valued function | ISI B.Stat Objective | TOMATO 690

Try this beautiful problem based on Real valued function, useful for ISI B.Stat Entrance

Real valued functions | ISI B.Stat TOMATO 690

Let \(f(x)\) be a real-valued function defined for all real numbers x such that \(|f(x) – f(y)|≤(1/2)|x – y|\) for all x, y. Then the number of points of intersection of the graph of \(y = f(x)\) and the line \(y = x\) is

  • 0
  • 1
  • 2
  • none of these

Key Concepts



Real valued function

Check the Answer

Answer: \(1\)

TOMATO, Problem 690

Challenges and Thrills in Pre College Mathematics

Try with Hints


\(|f(x) – f(y)| ≤ (1/2)|x – y|\)

\(\Rightarrow lim |{f(x) – f(y)}/(x – y)|\)( as x -> y ≤ lim (1/2)) as x - > y

\(\Rightarrow |f‟(y)| ≤ ½\)

\(\Rightarrow -1/2 ≤ f‟(y) ≤1/2\)

\(\Rightarrow -y/2 ≤ f(y) ≤ y/2\) (integrating)

\(\Rightarrow -x/2 ≤ f(x) ≤ x/2\)

Can you now finish the problem ..........

Real valued function graph

Therefore from the picture we can say that intersection point is \(1\)

Subscribe to Cheenta at Youtube

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.