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<p>We characterize the sets <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(E \subset \mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mo>⊂</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> such that there exists a continuous function <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(f: \mathbb {R}\rightarrow \mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\text{ Lip } f = \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.333333em" /> <mtext>Lip</mtext> <mspace width="0.333333em" /> <mi>f</mi> <mo>=</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> on <i>E</i> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\text{ lip } f = 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.333333em" /> <mtext>lip</mtext> <mspace width="0.333333em" /> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> on <i>E</i>.</p>

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Sets where \(\text{ Lip } f\) is infinite and \(\text{ lip } f\) vanishes.

  • 期刊论文

  • 发表日期：

    2025年10月18日

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  • Bruce Hanson

摘要

We characterize the sets \(E \subset \mathbb {R}\) E ⊂ R such that there exists a continuous function \(f: \mathbb {R}\rightarrow \mathbb {R}\) f : R → R such that \(\text{ Lip } f = \infty \) Lip f = ∞ on E and \(\text{ lip } f = 0\) lip f = 0 on E.