A Note on Discontinuous Functions with Continuous Second Iterate

Pingping Zhang¹, Xiaobing Gong²

¹Department of Mathematics, Binzhou University, Shandong, China.
²Department of Mathematics, Neijiang Normal University, Sichuan, China.

DOI: 10.4236/jamp.2015.36082                

Abstract

This paper investigates four classes of functions with a single discontinuous point. We give the sufficient and necessary conditions under which the second order iterates are continuous functions. Furthermore, the sufficient conditions for the continuity of the even order iterates with finitely many discontinuous points are obtained.

Keywords

Iteration, Discontinuous Point, Continuous Function

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Lesmoir-Gordon, N. and Edney, W. (2006) Introducing Fractal Geometry. Icon Books, Cambridge.
[2]	Lesmoir-Gordon, N. (2010) The Colours of Infinity: The Beauty and Power of Fractals. Springer, London. http://dx.doi.org/10.1007/978-1-84996-486-9
[3]	Mandelbrot, B. (2004) Fractals and Chaos: The Mandelbrot Set and Beyond. Springer, New York. http://dx.doi.org/10.1007/978-1-4757-4017-2
[4]	Zhang, P.P. (2012) Formal and Analytic Solutions for a Quadric Iterative Functional Equation. Electronic Journal of Differential Equations, 46, 1-9.
[5]	Zhang, P. and Gong, X. (2014) Existence of Solutions for Iterative Differential Equaitons. Electronic Journal of Differential Equations, 7, 1-10.
[6]	Zhang, P. and Mi, L. (2009) Analytic Solutions of a Second Order Iterative Functional Differential Equation. Applied Mathematics and Computation, 210, 277-283. http://dx.doi.org/10.1016/j.amc.2008.12.007
[7]	Zhang, P. and Zhang, Q. (2010) Local Invertible Analytic Solutions of a First Order Iterative Functional Differential Equation. Acta Mathematica Sinica Chinese Series, 53, 409-416.
[8]	Sun, D. (2004) Iteration of Quasi-Polynomial of Degree Two (in Chinese). J. Math., 24, 237-240.
[9]	Targonski, G. (1981) Topics in Iterative Theory. Vandenhoeck and Ruprecht, Götingen.
[10]	Wu, Z. and Sun, D. (2006) The Iteration of Quasi-Polynomials mappings (in Chinese). Acta Math. Sci. A., 26, 493-497.
[11]	Xu, L. and Xu, S. (2006) On Iteratio of Linear Fractional Function and Applications (in Chinese). Math. Pract. Theory., 36, 225-228.
[12]	Yu, Z., Yang, L. and Zhang, W. (2012) Discussion on Polynomials Having Polynomial Iterative Roots. Journal of Symbolic Computation, 47, 1154-1162. http://dx.doi.org/10.1016/j.jsc.2011.12.038
[13]	Liu, X., Liu, L. and Zhang, W. Discontinuous Function with Continuous Second Iterate. Aequat. Math.