Real Analysis Exchange

On Extendable Derivations

Tomasz Natkaniec

Full-text: Open access


There are derivations $f : \mathbb{R} \to \mathbb{R}$ which are almost continuous in the sense of Stallings but not extendable. Every derivation $f : \mathbb{R} \to \mathbb{R}$ can be expressed as the sum of two extendable derivations, as the discrete limit of a sequence of extendable derivations and as the limit of a transfinite sequence of extendable derivations. Analogous results hold for additive functions.

Article information

Real Anal. Exchange Volume 34, Number 1 (2008), 207-214.

First available in Project Euclid: 19 May 2009

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27}
Secondary: 54C08: Weak and generalized continuity

additive function derivation almost continuity extendability algebraically independent sets


Natkaniec, Tomasz. On Extendable Derivations. Real Anal. Exchange 34 (2008), no. 1, 207--214.

Export citation


  • D. Banaszewski, On some subclasses of additive functions, Ph.D. Thesis, Łódź University, 1997 (in Polish).
  • J. B. Brown, Negligible sets for real connectivity functions, Proc. Amer. Math. Soc., 24 (1970), 263-269.
  • K. Ciesielski, J. Jastrzębski, Darboux-like functions within the classes of Baire one, Baire two, and additive functions, Topology Appl., 103 (2000), 203–219.
  • R. Gibson, T. Natkaniec, Darboux like functions, Real Anal. Exchange, 22(2) (1996–97), 492–533.
  • R. G. Gibson and F. Roush, Connectivity functions with a perfect road, Real Anal. Exchange, 11 (1985–86), 260–264.
  • Z. Grande, On almost continuous additive functions, Math. Slovaca, 46 (1996), 203–211.
  • K. R. Kellum, Almost continuity and connectivity –- sometimes it's as easy to prove a stronger result, Real Anal. Exchange, 8 (1982–83), 244–252.
  • K. R. Kellum and B. D. Garret, Almost continuous real functions, Proc. Amer. Math. Soc., 33 (1972), 181–184.
  • M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Cauchy's Equation and Jensen's Inequality, PWN–Polish Scientific Publishers, Warszawa-Kraków-Katowice, 1985.
  • J. Mycielski, Independent sets in topological algebras, Fund. Math., 55 (1964), 139–147.
  • T. Natkaniec, Almost Continuity, Real Anal. Exchange, 17(2) (1991–92), 462–520.
  • H. Rosen, Limits and sums of extendable connectivity functions, Real Anal. Exchange, 20(1) (1994–95), 183–191.
  • W. Sierpiński, Sur les suites transfinies convergentes de fonctions de Baire, Fund. Math., 1 (1920), 132–141.
  • J. R. Stallings, Fixed point theorems for connectivity maps, Fund. Math., 47 (1959), 249-263.
  • E. Strońska, On almost continuous derivations, Real Anal. Exchange, 32(2) (2006–2007), 391–396.