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We prove that, almost surely, there exists no set $A\\subset[0,1]$ such that $\\dim A>\\frac12$ and $B\\colon A\\to\\mathbb{R}$ is $\\alpha$-H\\\"older continuous. The proof is an application of Kaufman's dimension doubling theorem. As a corollary of the above theorem, we show that, almost surely, there exists no set $A\\subset[0,1]$ such that $\\dim A>\\frac{\\beta}{2}$ and $B\\colon A\\to\\mathbb{R}$ has finite $\\beta$-variation. 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Let $\\alpha>\\frac12$ and $1\\leq \\beta \\leq 2$. We prove that, almost surely, there exists no set $A\\subset[0,1]$ such that $\\dim A>\\frac12$ and $B\\colon A\\to\\mathbb{R}$ is $\\alpha$-H\\\"older continuous. The proof is an application of Kaufman's dimension doubling theorem. As a corollary of the above theorem, we show that, almost surely, there exists no set $A\\subset[0,1]$ such that $\\dim A>\\frac{\\beta}{2}$ and $B\\colon A\\to\\mathbb{R}$ has finite $\\beta$-variation. 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