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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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