Let f : R R be such that f(x) - f(y) ≤ (x - y)^2 for all x, y ∈ R. Then f(1) - f(0) = ____ (Answer in integer)

GATE DA 2025 Q59: |f(x) − f(y)| ≤ (x − y)²: find f(1) − f(0)

Calculus and Optimization
2 marks
NAT
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Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be such that $\lvert f(x) - f(y) \rvert \le (x - y)^2$ for all $x, y \in \mathbb{R}$ . Then $f(1) - f(0) =$ ____ (Answer in integer)

Try it: numerical answer: type the value

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Numerical answer. No negative marking.

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Answer: 0 (official GATE 2025 final answer key)

Concept used: A bound of order $(x - y)^2$ forces the derivative to be zero everywhere.

First step: Divide by $\lvert x - y \rvert$ : $\left\lvert \dfrac{f(x) - f(y)}{x - y} \right\rvert \le \lvert x - y \rvert$ . Let $y \to x$ to see what $f'(x)$ must be.

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Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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GATE DA 2025 Q59: |f(x) − f(y)| ≤ (x − y)²: find f(1) − f(0)

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