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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

But try the problem first…

Answer: $1$

Source

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 $1$