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**

Limit

Calculas

Real valued function

## Check the Answer

But try the problem first…

Answer: \(1\)

TOMATO, Problem 690

Challenges and Thrills in Pre College Mathematics

## Try with Hints

First hint

Now,

\(|f(x) – f(y)| ≤ (1/2)|x – y|\)

\(\Rightarrow lim |{f(x) – f(y)}/(x – y)|\)( 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 ……….

Second Hint

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

## Other useful links

- https://www.cheenta.com/integers-and-divisors-isi-b-stat-entrance-tomato-98/
- https://www.youtube.com/watch?v=fRj9NuPGrLU&t=279s

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