## Real Analysis Exchange

### Continuity and Differentiability Aspects of Metric Preserving Functions

Robert W. Vallin

#### Abstract

A function $f$ is metric preserving if for every metric space $(M,\rho)$ we have that $f \circ \rho$ is still a metric on $M$. In this article we look at the behavior of such functions with respect to continuity and differentiability. We include several pathological examples and some open questions.

#### Article information

Source
Real Anal. Exchange, Volume 25, Number 2 (1999), 849-868.

Dates
First available in Project Euclid: 3 January 2009

https://projecteuclid.org/euclid.rae/1230995419

Mathematical Reviews number (MathSciNet)
MR1778537

Zentralblatt MATH identifier
1016.26004

#### Citation

Vallin, Robert W. Continuity and Differentiability Aspects of Metric Preserving Functions. Real Anal. Exchange 25 (1999), no. 2, 849--868. https://projecteuclid.org/euclid.rae/1230995419

#### References

• Borsík, J. and Doboš, J., On metric preserving functions, Real Analysis Exchange, 13(1987–88), 285–293.
• Borsík, J. and Doboš, J., On metrization of the uniformity of a product of metric spaces, Math. Slovaca, 32(1982), 97–102.
• Borsík, J. and Doboš, J., On a product of metric spaces, Math. Slovaca, 31(1981), 193–205.
• Borsík, J. and Doboš, J., Functions whose composition with every metric is a metric, Math. Slovaca, 31(1981), 3–12 (in Russian).
• Bruckner, A., Differentiation of real functions, 2nd Ed., Rhode Island, AMS, 1994.
• Corazza, Paul, Introduction to metric-preserving functions, Amer. Math. Monthly, 106(4), 309–323.
• Doboš, J., The standard Cantor function is subadditive, Proc. Amer. Math. Soc., 124(1996), 3425–3426.
• Doboš, J., On modifications of the Euclidean metric on the reals, Tatra Mountains Math. Publ., 8(1996), 51–54.
• Doboš, J., A survey of metric preserving functions, Questions and Answers in General Topology, Vol 13(1995), 129–134.
• Doboš, J., On a certain lattice of topologies on a product of metric spaces, Math. Slovaca, 32(1982), 397–402.
• Doboš, J. and Piotrowski, Z., Some remarks on metric preserving functions, Real Analysis Exchange, 19(1993–1994), 317–320.
• Doboš, J. and Piotrowski, Z., A note on metric preserving functions, Inter. J. Math and Math. Sci., 19(1996), 199–200.
• Doboš, J. and Piotrowski, Z., When distance means money, Int. J. Math. Educ. Sci. Technol., 28(1997), 513–518.
• J\ruza, M., A note on complete metric spaces, Matematicko–Fyzikálny Časopis SAV, VI, 3(1956), 143–148 (Czech).
• Kelley, J.L., General Topology, Van Nostrand, New York, 1955.
• Law, M., “Absolutely” metric preserving functions: functions that preserve the absolute value metric, Master's Project, the University of North Dakota, 1995.
• Piotrowski, Z., On integer-valued metrics, School of Math., Phys. Chem. Wroclaw Univ. (Poland), 1974 (Polish).
• Pokorný, I., Remarks on the sum of metrics, Tatra Mountains Math. Publ., 14(1998), 63–65.
• Pokorný, I., Some remarks on metric preserving functions of several variables, Tatra Mountains Math. Publ., Vol 8(1996), 89–92.
• Pokorný, I., Some remarks on metric preserving functions, Tatra Mountains Math. Publ., 2(1993), 66–68.
• Sreenivasan, T.K., Some properties of distance functions , J. Indian Math. Soc., 11(1947), 38–43.
• Terpe, F., Some properties of metric preserving functions, Proc. Conf. Topology, Measure, and Fractals, Math. Res. 66, Akademie-Verlag, Berlin, (1992), 214–217.
• Terpe, F., Metric preserving functions of several variables, Proc. Conf. Topology and Measure V, Greifswald (1988), 169–174.
• Vallin, R.W., On preserving $( \mathbb{R} ,Eucl.)$ and almost periodic functions, (submitted).
• Vallin, R.W., On metric preserving functions and infinite derivatives, Acta Math. Univ. Comenianae, Vol 67, No.2(1998), 373–376.
• Vallin, R.W., A subset of metric preserving functions, Int. J. Math. and Math. Sci., Vol. 21, No. 2, 409–410.
• Watson, S., Review of A survey of metric preserving functions, Mathematical Reviews, AMS, 96m.
• Wilson, W.A., On certain types of continuous transformations of metric spaces, Amer. J. Math. 57(1935), 62–68.