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

GATE 2025 · Engineering Mathematics · Limits · medium

Answer: 0

1. Symmetrise the inequality: Combining: -(x-y)^2 <= f(x)-f(y) <= (x-y)^2, i.e. |f(x)-f(y)| <= (x-y)^2.
2. Show derivative is zero: For any x, |f'(x)| = lim_{y->x} |f(x)-f(y)|/|x-y| <= lim_{y->x} |x-y| = 0. So f'(x) = 0.
3. Conclude f is constant: Since f has zero derivative everywhere on R, f(1) = f(0), so f(1) - f(0) = 0.