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Note on functions satisfying the integral Hölder condition. (English) Zbl 0863.26006
For $$q\geq 1$$ and a 1-periodic function $$f:\mathbb R \to \mathbb R\cup \{-\infty,\infty\}$$ define $$|f|_q=[\int_0^1|f(x)|^q\text{d}x]^{1/q}$$ and in the case $$|f|_q<\infty$$ its modulus of continuity $$\omega(f,t)_q=\sup_{|h|\leq t}[\int_0^1|f(x+h) - f(x)|^q\text{d}x]^{1/q}$$. Two such functions are equivalent if they differ almost everywhere by a constant. Given a modulus of continuity $$\omega$$, $$H^\omega_q$$ is the set of classes of equivalent functions $$f$$ for which there is a nonnegative constant $$c$$ such that $$\omega(f,t)_q\leq c\omega(t)$$. It is shown that if $$\liminf_{t\to 0}\omega(t)/t=\infty$$ then the set $$\{f\in H^\omega_q:\text{var}(f) <\infty\}$$ is of the first category in $$H^\omega_q$$ and that if $$\liminf_{t\to 0}\omega(t)/t<\infty$$ then $$\{f\in H^\omega_q:\text{var}(f) <\infty\}=H^\omega_q$$.
##### MSC:
 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable 26A45 Functions of bounded variation, generalizations
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