Wednesday, February 26, 2014

Exceptions to Continuity 2

This is the second post of a three-part post. For the first, see here.

The question arose in the previous post whether for any set S in R, the real numbers, there exists a function f such that S is the set of discontinuities of f, or, in our previous notation, D(f) = S. Note that this problem is exactly equivalent to the problem of whether every set can be the set of points at which a function is continuous, since every function f is continuous at exactly the points that it is not discontinuous. Before tackling the whole problem, we solve it for a more limited class of functions, called the monotone functions.

A monotonically increasing function on an interval (image from wikipedia)
A function is monotone if it is either always increasing or always decreasing (such a function is called a monotonically increasing or monotonically decreasing function). A function f is increasing if for any real numbers a and b, ab means that f(a) ≤ f(b), with a similar definition for decreasing. The function above increases, then stays constant for awhile, and then continues to increase. It is thus a monotonically increasing function. Some common examples of monotonically increasing functions are y = x, y = c (for any constant c), y = xn (where n is an odd integer), and y = ex.

Next, it is important to note that there is a one-to-one correspondence between monotonically increasing and monotonically decreasing functions caused by flipping sign, that is, the negative of a monotonically increasing function is monotonically decreasing, and the negative of every monotonically decreasing function is monotonically increasing. Thus y = -x, etc., are monotonically decreasing.

A theorem in mathematics called Froda's theorem after the Romanian mathematician Alexandru Froda completely classifies the possible sets of discontinuities over the monotone functions. We outline the proof of the theorem, noting that, due to the one-to-one correspondence discussed above, the result for monotonically increasing functions will hold for all monotone functions. The statement runs thus: For any monotone function f defined on closed interval [a,b], D(f) is at most a countable set in [a,b], that is, the set is empty (for an everywhere continuous function), finite, or countable (can be put in a one-to-one correspondence with the natural numbers).

The first step in the proof is noting that, for a monotonically increasing function f defined on the interval [a,b], the only discontinuities that f can have are jump discontinuities. Other types of continuities require some type of infinite oscillation (e.g. the function y = sin(1/x)), but this involves the function increasing and decreasing, which is not allowed for monotone functions. Thus we consider jump discontinuities only.

Consider the difference d = f(b) - f(a) of the values of the function at the endpoints of the interval [a,b]. Clearly it is positive and finite, due to the properties of f and the fact that it is defined on both endpoints. Note also that the jumps can only be positive, since the function is increasing; the function can only jump up, not down. For any positive number ε, therefore, there can only be a finite number of jumps of length ε, since the sum of the jumps cannot exceed d.

For example, in the above diagram, the function f, which increases from 1 to 3 over the interval [a,b], cannot have more than two jump discontinuities with jump 1, since the sum would then exceed the value of d, in this case, 2.

Therefore, given a monotonically increasing function f, for each positive number ε we can construct a set of points {x1,x2,...,xn}, containing every point that is a jump discontinuity of f with a jump greater than or equal to ε. By the previous remarks, this set is finite for any positive ε (although it is allowed to be empty, e.g., for continuous monotonically increasing functions). Finally, we define Dn(f) for a positive integer n to be the set corresponding to the value ε = 1/n. The union of all of the Dn for positive integer n then includes every jump discontinuity of f, since for any jump discontinuity with jump ε, ε is greater than 1/n for some n. If the jump were 0, the discontinuity would not be a discontinuity at all! Thus every jump discontinuity is in some Dn, and thus in their union, which we shall call D, as consistent with previous notation.

Therefore, the set of discontinuities of a monotonically increasing function f (and similarly for all monotone functions) is a countable union of finite sets, which is countable.

This result is extensible to a monotone function on the real numbers. The set of real numbers R can be expressed as a countable union of closed intervals [n,n + 1], where the union is over every integer n. One can simply collect the discontinuities from each interval (on which the function must be monotone, since it is monotone everywhere) into a larger set. Since a countable union of countable sets is countable, the set of discontinuities over all of R is countable also. Lastly, this result also extends to any function f defined on the real numbers such that there is a countable collection of sets A such that the union of all members of A is R and f is monotone on each member of A. In other words, if the real numbers can be appropriately split into pieces on which a function f is monotone, then the set of discontinuities of f is countable.

As general as this result may be, it certainly does not classify the sets of discontinuities of all functions. We have already seen an example of a function discontinuous everywhere. The function of that example is not monotone on any interval, so Froda's theorem is inapplicable.

To complete the classification of sets of discontinuities, we must introduce the notion of a Fσ set. A subset S of the real numbers is an Fσ set if and only if it is a countable union of closed sets. A closed set in the context of the real numbers is one which contains its boundary points. For example, [0,1], the interval containing 0 and 1, is a closed interval, because it contains its boundary points, 0 and 1. The sets (0,1), [0,1), and (0,1], missing one or both endpoints, are not closed. In addition, any union of disjoint closed intervals such as [-1,2]∪[3,4]∪[5,7] is closed.

The realm of Fσ sets is quite general. First, since R itself, the set of real numbers, has no boundary, it by definition must "contain" its boundary, and is thus closed, and therefore Fσ. Since the empty set has no boundary, it is also closed, and {} is Fσ. Any countable set of points is also Fσ, as a set containing one point, {x}, is always closed, and the collection of Fσ sets is closed under countable union. Thus Q is Fσ. Finally, as an example of a set that is not Fσ, consider the set of irrationals, R - Q. Since Q is dense, i.e. there is a rational number between any two distinct real numbers, no part of R - Q has an interior; there are no "blocks" of irrational numbers with no intervening rationals among the reals. Thus to get the irrationals into a collection of closed sets, we have to "package" them individually, using, for example, countable collections of points. However, the irrationals are uncountable, so no countable union of closed sets can yield R - Q. With this definition, we state the following theorem completely classifying sets of discontinuities:

A set S can occur as a set of discontinuities of a function f on the real numbers if and only if S is Fσ.

The proof that every function f has D(f) an Fσ set is too difficult to consider here. However, we can demonstrate the proof in the reverse direction. To finish this series (see the next post, coming March 6), we construct a function whose discontinuities form an arbitrarily chosen Fσ set.

Sources: Counterexamples in Analysis by Bernard R. Gelbaum and John M. H. Olmsted, Froda's Theorem on Wikipedia,,,

No comments: