Opuscula Math. 31, no. 1 (2011), 75-96
http://dx.doi.org/10.7494/OpMath.2011.31.1.75

Opuscula Mathematica

Janusz Matkowski

Abstract. Some conditions under which any subadditive function is periodic are presented. It is shown that the boundedness from below in a neighborhood of a point of a subadditive periodic (s.p.) function implies its nonnegativity, and the boundedness from above in a neighborhood of a point implies it nonnegativity and global boundedness from above. A necessary and sufficient condition for existence of a subadditive periodic extension of a function $$f_{0}:[0,1)\rightarrow \mathbb{R}$$ is given. The continuity, differentiability of a s.p. function is discussed, and an example of a continuous nowhere differentiable s.p. function is presented. The functions which are the sums of linear functions and s.p. functions are characterized. The refinements of some known results on the continuity of subadditive functions are presented.

Keywords: subadditive function, periodic function, periodic extension, concave function, continuity, continuous nowhere differentiable function.

Mathematics Subject Classification: 39B62, 26A51.

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• Janusz Matkowski
• University of Zielona Góra, Faculty of Mathematics, Computer Science and Econometry, ul. Szafrana 4a, 65-516 Zielona Góra, Poland
• Silesian University, Institute of Mathematics, ul. Bankowa 14, 40-007 Katowice, Poland
• Revised: 2010-05-31.
• Accepted: 2010-07-12.