Almost continuous real functions

A blocking set of a function j is a closed set which does not intersect f but which intersects each continuous function with domain the same asf. It is shown that for each function which is not almost continuous, there exists a minimal blocking set. Using this property it is shown that there exists an almost continuous function with domain [0, 1] which is a G5 set but is not of Baire Class 1, and that there exists an almost continuous function dense in the unit square. Unless otherwise stated, all functions considered are real functions with domain a closed and bounded subset of the real line, R. No distinction is made between a function and its graph. If the functionf is a connected subset of the plane, f is simply said to be a connected function. If each open set containingf also contains a continuous function with the same domain as f, then the function f is said to be almost continuous. Stallings [7] showed that if the functionf is almost continuous and has connected domain, then f is connected. He stated as an open question the following. Is each connected function almost continuous? This question was answered in the negative by several authors ([1], [2], [4], [6]). In [4], Jones and Thomas showed that there exists a function which is not of Baire Class 1, is a G5 set, is connected but is not almost continuous. In Example 1 of the present paper, it is shown that there exists such a function which is almost continuous. Example 2 shows that there exists an almost continuous function which is a dense subset of the unit square. In [1], Brown gave an example of a connected but not almost continuous function dense in [0, 1] x R. Suppose M is a subset of the plane. The X-projection of M is denoted by (M)x. The Y-projection of M is denoted by (M)y. If Kc (M)x, MK denotes the part of M with X-projection K. The function f is said to be of Baire Class 1 if f is the pointwise limit of a sequence of continuous functions. Suppose that D is a set containing the functionf The statement that the subset C of D is a blocking set off relative to D means that C is closed Received by the editors June 15, 1971. AMS 1970 subject classifications. Primary 54C10, 54C30, 26A15; Secondary 26A21.