## Real Analysis Exchange

### Continuous and Smooth Images of Sets

#### Abstract

This note shows that if a subset $S$ of $\mathbb{R}$ is such that some continuous function $f\colon\mathbb{R}\to\mathbb{R}$ has the property “$f[S]$ contains a perfect set,” then some $\mathcal{C}\,^\infty$ function $g\colon\mathbb{R}\to\mathbb{R}$ has the same property. Moreover, if $f[S]$ is nowhere dense, then the $g$ can have the stronger property “$g[S]$ is perfect.” The last result is used to show that it is consistent with ZFC (the usual axioms of set theory) that for each subset $S$ of $\mathbb{R}$ of cardinality $\mathfrak{c}$ (the cardinality of the continuum) there exists a $\mathcal{C}\,^\infty$ function $g\colon \mathbb{R}\to\mathbb{R}$ such that $g[S]$ contains a perfect set.

#### Article information

Source
Real Anal. Exchange, Volume 37, Number 2 (2011), 305-314.

Dates
First available in Project Euclid: 15 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.rae/1366030626

Mathematical Reviews number (MathSciNet)
MR3080593

Zentralblatt MATH identifier
1283.26004

#### Citation

Ciesielski, Krzysztof Chris; Nishiura, Togo. Continuous and Smooth Images of Sets. Real Anal. Exchange 37 (2011), no. 2, 305--314. https://projecteuclid.org/euclid.rae/1366030626

#### References

• A.M. Bruckner, Differentiability of real functions, Lecture Notes in Mathematics Vol. 659, Springer-Verlag 1978.
• K. Ciesielski, J. Pawlikowski, Covering Property Axiom CPA. A combinatorial core of the iterated perfect set model, Cambridge Tracts in Mathematics 164, Cambridge Univ. Press, 2004.
• K. Ciesielski and S. Shelah, A model with no magic set, J. Symbolic Logic, 64(4) (1999), 1467–1490.
• P. Corazza, The generalized Borel conjecture and strongly proper orders, Trans. Amer. Math. Soc., 316(1) (1989), 115–140.
• J. Foran, Fundamentals of Real Analysis, Marcel Dekker, New York-Basel-Hong Kong, 1991.
• C. Goffman, T. Nishiura, and D. Waterman, Homeomorphisms in analysis, Mathematical Surveys and Monographs vol 54, American Mathematical Society 1997.
• A.W. Miller, Mapping a set of reals onto the reals, J. Symbolic Logic 48, (1983), 575–584.
• A.W. Miller, Special Subsets of the Real Line, in Handbook of Set-Theoretic Topology (K. Kunen and J.E. Vaughan, eds.), North-Holland (1984), 201–233.