Consider the functions from onto . By a well known theorem of Peano from 1890 (see e.g. ) such an F can be continuous. However, it is not difficult to see that it cannot be differentiable. It follows easily from the fact that every differentiable function satisfies the Banach condition , i.e., the set has Lebesgue measure zero. (See e.g. [118, Chap. VII, p. 221,].) Thus, Morayne in 1987 considered the following question: can function be chosen in such a way that at every point either or is differentiable? The surprising answer is given below.
The proof of this theorem is based on a well known theorem of Sierpinski [127, Property ,] from 1919 that CH is equivalent to the existence of a decomposition of into two sets A and B such that all horizontal section of A and all vertical section of B are at most countable. It is also worthwhile to point out that the function F from Theorem 3.1 is not a Peano curve, since it is not continuous. In fact Morayne proves in the same paper that for such an it is impossible that even one of or is measurable.
Next, recall that if two continuous functions agree on
some dense set then they are equal. Does the statement remain
true if the clause ``agree on M'' is replaced by ``f[M]=g[M]?'' Clearly not,
as shown by and any two different rational translations of the
identity function. What about finding some more complicated set
for which the implication
holds for any continuous f and g? Even this is too much to ask, as recently noted by Burke and Ciesielski [18, Remark 6.6,]. On the other hand, the following theorem of Berarducci and Dikranjan from 1993 gives a positive (consistent) answer to this question in the class of continuous nowhere constant functions. (Function is nowhere constant if it is not constant on any non-empty open set.)
The construction of a magic set, given in , is done by an easy diagonal transfinite induction argument, and uses only the assumption that less than continuum many meager sets do not cover . In particular, CH can be replaced by MA in Theorem 3.2.
Examining the problem of existence of a magic set in ZFC Burke and Ciesielski noticed the following properties of a magic set.
In fact part (b) of Theorem 3.3 is just a remark: if there were a continuous with then it could be easily modified to a nowhere constant function such that , and the functions f and g=1+f would give a contradiction. But (b) shows that there is no magic set of cardinality continuum in the model from Theorem 2.4, the iterated perfect set model. Although it was noticed in  that in this model there exists a magic set (clearly of cardinality less than ), Theorem 3.3 was used by Ciesielski and Shelah as a base in proving that magic set cannot be constructed in ZFC.
The magic sets for different classes of functions have also been considered. Burke and Ciesielski  studied such sets (which they call sets of range uniqueness) for the classes of measurable functions with respect to abstract measurable spaces with negligibles. In particular, they proved the following theorem concerning the Lebesgue measurable functions.
The model satisfying Theorem 3.5(b) is a modification the iterated perfect set model and was constructed by Corazza  in 1989. Once again it satisfies property (b) of Theorem 3.4, while part (a) is replaced by . It has been also proved by Ciesielski and Larson that for the class of functions (continuously differentiable) the existence of a magic set can be proved in ZFC.
For the following consideration recall that a function is Darboux (or has the Darboux property) if f[C] is connected for every connected subset C of . Thus, in case of n=1 Darboux functions are precisely the functions for which the Intermediate Value Theorem holds. The class of Darboux functions will be denoted here by (with n clear from the context, usually n=1).
The class of Darboux functions has been studied for a long time as one of possible generalizations of the class of continuous functions. (Clearly every continuous function is Darboux.) However, it has some peculiar properties. For example, it is not closed under addition. In fact, in 1927 Lindenbaum  noticed (without a proof) that every function can be written as a sum of two Darboux functions. (For proofs, see [129, ].) This theorem has been improved in several ways. Erdos  showed that if f is measurable, both of the summands can be chosen to be measurable. Another improvement was done by Fast  in 1974 who proved that for every family of real functions that has cardinality continuum there is just one Darboux function g such that the sum of g with any function in has the Darboux property. The natural question of whether such a ``universal'' summand exists also for families of larger cardinality has been studied by Natkaniec  and lead to the development described in Section 4.
A problem that is in some sense opposite to the existence of a ``universal'' summand is for which families of functions there is a ``universally bad'' Darboux function g, in the sense that the sum of g with any function in does not have the Darboux property. In 1990 Kirchheim and Natkaniec addressed this problem for the class of continuous nowhere constant functions.
The problem whether the additional set-theoretic assumptions are necessary in this theorem was investigated in 1992 by Komjáth  and was settled in 1995 by Steprans.
A model having this property is the iterated perfect set model. Note also that in Theorem 3.7 the restriction to the nowhere constant functions is important. This has been proved independently by T. Natkaniec (in his 1992/93 paper ) and by J. Steprans (in the 1995 paper mentioned above).
To state farther results recall the following generalizations of continuity. A function is almost continuous (in sense of Stallings) if each open subset of containing the graph of f contains also a continuous function from to . Function has a perfect road at if there exists a perfect set C such that x is a bilateral limit point of C and is continuous at x . The classes of all almost continuous functions and all functions having a perfect road at each point are denoted by and , respectively. It is easy to see that (for functions on ) and that the inclusions are strict (see e.g. ), where stands for the class of all continuous functions. We will also consider the class of Sierpinski-Zygmund (SZ-) functions, i.e., functions whose restrictions are discontinuous for all subsets X of of cardinality continuum. (That is, functions from Theorem 2.6.)
The classes and recently appeared in a 1993 paper of Darji , who constructed in ZFC a function . Answering a question posed by Darji, this year Balcerzak, Ciesielski and Natkaniec proved the following theorem.
The model satisfying Theorem 3.10(b) is, once again, the iterated perfect set model.
Another generalization of continuity is that of countable continuity: a function is is countably continuous if there exists a countable partition of such that the restriction of f to any is continuous. (See also Section 4.) In 1995 Darji gave the following combinatorial characterization of this notion.
The characterization () cannot be proved in ZFC. This follows from a result of Cichon and Morayne  from 1988 which implies that in some models of ZFC (actually, when and , where d is the dominating number) () is false. However, it is not known, whether the equivalence () can be proved in absence of CH, leading to the following open problem.
Another recent theorem concerning countable and symmetric continuities is the following theorem of Ciesielski and Szyszkowski, answering a question of L. Larson.
We will finish this section with the following two interesting results. The first one has been proved independently in 1978 by Grande and Lipinski and in 1979 by Kharazishvili.
This theorem has important consequences concerning the existence of solutions of the differential equation in the class of absolutely continuous functions. In 1992 Balcerzak  showed that in Theorem 3.13 the CH assumption can be weakened to . However, the following problem remains open.
The second result is the following 1974 theorem of R. O. Davies.
Note that Theorem 3.14 is related to Hilbert's Problem 13 (from his famous Paris lecture of 1900) and a 1957 theorem of Kolmogorov, in which he proves that every continuous function can be represented in a certain form (similar to the above) by continuous functions of one variable. An interesting account on this and related results can be found in a 1984 paper of Spreceher .