Try this beautiful problem based on Real valued function, useful for ISI B.Stat Entrance
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
Limit
Calculas
Real valued function
Answer: \(1\)
TOMATO, Problem 690
Challenges and Thrills in Pre College Mathematics
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 ..........
Therefore from the picture we can say that intersection point is \(1\)
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