ON SOME PROPERTIES OF THE SPACES OF ALMOST CONTINUOUS FUNCTIONS

In this paper we prove that, in the space 𝒜 of almost continuous functions (with the metric of uniform convergence), the set of functions of the first class of Baire is superporous at each point of this space


R.J. PAWLAK
We say that a function f X )", where X and Y are arbitrary topological spaces, is almost continuous if, for any open set V C X x Y containing the graph G(f), V contains the graph of some continuous function g X The set of all almost continuous fitnctions f I I and the metric space consisting of these functions with the metric " of ulit'orm convergence is denoted by the symbol 4. By the letter .A we denote the set of all f A which are functions of the first class of Baire.
Suppose f X Y.The statement that the subset of X x Y is a blocking set of f in X x Y means that C is closed relative to X x Y, C t3 (f) q) and C intarsects 7(g) whenever g" X Y is a continuous function.If no proper subset of C is a blocking set of f in X x Y, C is said to be the minimal blocking set of f in X x Y.
Let A C 12. Then projx(A) denotes the projection of the set A onto the X-axis and, in the case when A is a closed set, we assume (A)r,, {(z0, Y0) A" y0 min{y" (xo, y) A}}.
Let f" I I.If A C I (f(I) C B), then by the symbol flA (flls) we understand a function from A to I (from I to B as a subspace of I).
For 0 < ( < < 1, we define a function f' I I by letting: Let X be a metric space.The open ball with centre x and radius r > 0 will be denoted by K(z, r).Let M C X, x X and R > 0. Then we denote by 7(z,R, M) the supremum of the set of all r > 0 for which there exists z 5 X such that K(z,r) C K(x,R) \ M. The number p(M,x) 2-limsups_.0+(,s,M)t is called the porosity of M at z ( [10]).M is porous at x if p(M, x) > 0. We say that E C X is superporous at z X if E U F is porous at z whenever E is porous at z ( [11]).Now we formulate three lemmas showing some properties of almost continuous functions.These properties are applied later.
Lemma 1 Let f: I --, I ba a function such that f I) C [cr,/31 where < .T hen f .A if and only if fllI,,al is almost continuous.
Remark 1 If f X --, Y, where X, Y are toplogical spaces, is a continuous function such that f(X) C B C Y, then flls is a continuous function, too.However it is not diJflcult to construct an example of a topology T defined in I and an almost continuous function I (1, 7r) such that 9(I)= [1/2,1] and gilt1/2,1] /s not almost continuous.Lemma 2 ([6]) A function f: I I is almost continuous if and only if flt,bl is almost contin- uous for any a, b [0,1].Remark 2 It is known that there exists an almost continuous real function defined on some compact subset K of the plane, such that the restriction of this function to the set Int K is not almost continuous( [6]).Lemma 3 Let f: I [, 3] be a function for which there exists an interval (a, b) C 1 such that fll\(a,li and fll\t0,b) ave almost continuous and f is not almost continuous.Then projx(h'l)fq(a, b) is a nondegenerate interval, where KI C I [a, 3] is the minimal blocking set of f.Proof.It is suiticicnt to prove (see, for exavnple, [3], Theorem 1; [9], Lemma 4; [6], Theorem 1.1.2) that projx(Kl)f3 (a.b) .A ssume, to the contrary, that prodx(Kl) (o,b) 0. Thus, according to the above-cited theor('ns, either I( C {(x,y) x a} or lf C {(x,y) x b}.Without loss of generality assume t}at Kl C {(x.y) x a}.Of corse, K f G(f) 0, and so, Kl9(f*) wherc f" .flt(,,].L('t us considcr the set A ((I (a. 1])x [o,]) KI.Thus   A is a neighbourhood of (.f')(in (1 (a, 1]) x [&,/]).This means that A cow,talus the graph of some continuous function g*: I (a,l] [a, fl].Let (for x G I) 9*(x) if xEl\(a,l], g*(a) f x >_ a.
Then g is continuous and K I gl (g) q), which is impossible because K! is the blocking set of f.The contradictions obtained ends the proof of the lemma.Theorem 1 The set 4 is perfect and superporous at each point of the space .A.
Proof.According to J.B. Brown's theorem and the well-known fact (see, for example, [2], Theorem 2.3.4]) that the limit of a uniformly convergent sequence of Darboux functions of the first class of Baire is a Darboux function of the first class of Baire, we deduce that A1 is closed in ,4.
In this way the fact that .A is perfect has been demonstrated.Now, we shall show that 41 is superporous at each point of the space .A.
So, let M be an arbitrary porous set at t. Assume the notation from the definition of a porous set.Let p(M,t) 2a > 0. This means that there exists a sequence {R,} such that P " 0 and lim 7(t, R,,, M) a.
Let n be a fixed number.From the definition of 7(t, R,, M) it follows that (for n sufficiently large) there are d A and , >_ 7(t,R,,M)-.R,, > 0, such that K(d,,) C K(t,t) \ M.