How can I prove that $f(x)=x^s$ for $0< s ≤1$ is Hölder-continuous with constant s?
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1What have you tried? – DMcMor Jan 14 '21 at 17:51
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3Does this answer your question? proving that $f(x) = x^s$ is holder continuous with holder exponent s – oliverjones Jan 14 '21 at 18:07
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The proofs in that link do not seem really convincing. I want to show that the constant of Hölder continuity is s. – Lali Loli Jan 14 '21 at 18:54
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I started by assuming that x>y and then I tried to use difference of powers formula but it doesn't work here. I can neither use limits of two variable functions. – Lali Loli Jan 14 '21 at 18:58
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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 $[x,\infty).$
zhw.
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