GATE 2025 DA – Question 59
Let $f: \mathbb{R} \to \mathbb{R}$ be such that $|f(x) - f(y)| \le (x - y)^2$ for all $x, y \in \mathbb{R}$. Then $f(1) - f(0) = $ ______ (*Answer in integer*)
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Correct answer: 0
Explanation
Divide by $|x - y|$: $\left|\frac{f(x) - f(y)}{x - y}\right| \le |x - y|$. Letting $y \to x$ shows $f'(x) = 0$ at every $x$, so $f$ is constant. Hence $f(1) - f(0) = 0$.