How can I prove that $f(x)=x^s$ for $0< s ≤1$ is Hölder-continuous with constant s?
2026-03-29 11:51:43.1774785103
Prove Hölder continuity of power function
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Hint: The result is true for $s=1,$ so assume $0<s<1.$ Suppose $0\le x.$ We'll be done if we show $y^s-x^s \le (y-x)^s$ for $y\ge x.$ Define $f(y) = y^s-x^s - (y-x)^s.$ Show $f'(y) < 0$ on $(x,\infty)$ to see $f$ decreases on $[x,\infty).$