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A class of subadditively continuous real functions. (English) Zbl 0358.26003
The author defines \(f \colon \mathbb{R} \to \mathbb{R}\) to be subadditively continuous (s.c.) at \(x\) if and only if \(\forall \varepsilon >0\) \(\exists\delta>0\) such that \(|f(y)|<\delta \Rightarrow |f(x\pm y) - f(x)|< \varepsilon\); concepts such as s.c. (i.e. \(\forall x\)), uniform s.c., left and right s.c. are also introduced. The following results are proved: (i) if \(f(0)\neq 0\) then \(f\) is s.c. if and only if \(\exists \varepsilon >0\) s.t. \(|f|\geq \varepsilon\); (ii) if \(x\) is a point of continuity of \(f\) and \(f(x)\neq 0\) and \(0\) is not a limit value of \(f\) at \(x\) then \(f\) is s.c. at \(x\); (iii) if \(y\rightarrow\) iff \(f(y)\rightarrow 0\) then \(f\) is s.c. at \(x\) if and only if it is continuous at \(x\); (iv) if \(f\) is s.c., and either \(0= f(y)\) or \(0\) is a limit value of \(f\) at \(y\), \(y\neq 0\), then if \(f\) is continuous at \(x\), \(f(x)=f(x\pm y)\). Other results connecting these concepts with periodicity and almost periodicity are also proved.
Reviewer: P. S. Bullen
MSC:
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
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References:
[1] DOBRAKOV I., FARKOVÁ J.: On submeasures II. · Zbl 0428.28001
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