The level structure of a residual set of continuous functions

Authors:
A. M. Bruckner and K. M. Garg

Journal:
Trans. Amer. Math. Soc. **232** (1977), 307-321

MSC:
Primary 26A27; Secondary 26A48, 46E15

DOI:
https://doi.org/10.1090/S0002-9947-1977-0476939-X

MathSciNet review:
0476939

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let *C* denote the Banach space of continuous real-valued functions on with the uniform norm. The present article is devoted to the structure of the sets in which the graphs of a residual set of functions in *C* intersect with different straight lines.

It is proved that there exists a residual set *A* in *C* such that, for every function , the top and the bottom (horizontal) levels of *f* are singletons, in between these two levels there are countably many levels of *f* that consist of a nonempty perfect set together with a single isolated point, and the remaining levels of *f* are all perfect. Moreover, the levels containing an isolated point correspond to a dense set of heights between the minimum and the maximum values assumed by the function.

As for the levels in different directions, there exists a residual set *B* in *C* such that, for every function , the structure of the levels of *f* is the same as above in all but a countable dense set of directions, and in each of the exceptional nonvertical directions the level structure of *f* is the same but for the fact that one (and only one) of the levels has two isolated points in place of one. For a general function a theorem is proved establishing the existence of singleton levels of *f*, and of the levels of *f* that contain isolated points.

**[1]**S. Banach,*Über die Bairesche Kategorie gewisser Funktionenmengen*, Studia Math.**3**(1931), 174-179.**[2]**K. M. Garg,*On nowhere monotone functions. I. Derivatives at a residual set*, Ann. Univ. Sci. Budapest. Eötvös Sect. Math.**5**(1962), 173–177. MR**146317****[3]**K. M. Garg,*On nowhere monotone functions. III. (Functions of first and second species)*, Rev. Math. Pures Appl.**8**(1963), 83–90. MR**151560****[4]**Krishna Murari Garg,*On a residual set of continuous functions*, Czechoslovak Math. J.**20 (95)**(1970), 537–543. MR**0268334****[5]**K. M. Garg,*On bilateral derivates and the derivative*, Trans. Amer. Math. Soc.**210**(1975), 295–329. MR**369629**, https://doi.org/10.1090/S0002-9947-1975-0369629-3**[6]**J. Gillis,*Note on a conjecture of Erdös*, Quart. J. Math. Oxford Ser.**10**(1939), 151-154.**[7]**E. W. Hobson,*The theory of functions of a real variable and the theory of Fourier's series*, Vol. II, Dover, New York, 1958. MR**19**#1166.**[8]**V. Jarnik,*Über die Differenzierbarkeit stetiger Funktionen*, Fund. Math.**21**(1933), 48-58.**[9]**S. Mazurkiewicz,*Sur les fonctions non dèrivables*, Studia Math.**3**(1931), 91-94.**[10]**S. Saks,*On the functions of Besicovitch in the space of continuous functions*, Fund. Math.**19**(1932), 211-219.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
26A27,
26A48,
46E15

Retrieve articles in all journals with MSC: 26A27, 26A48, 46E15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1977-0476939-X

Keywords:
Banach space ,
graph of continuous functions,
structure of level sets,
perfect levels,
derivates,
nondifferentiable functions

Article copyright:
© Copyright 1977
American Mathematical Society