Question

Let \(f\) be a real function defined on \((a, b)\). Prove that the set of points at which \(f\) has a simple discontinuity is at most countable. Hint: Let \(E\) be the set on which \(f(x-)p\). The set of all such triples is countable. Show that each triple is associated with at most one point of \(E\). Deal similarly with the other possible types of simple discontinuities.

Short answer

Question: Prove that the set of points at which a real function \(f\) has a simple discontinuity on the open interval \((a, b)\) is at most countable. Answer: We have shown that the set of points at which the function \(f\) has a simple discontinuity can be divided into two types: jump discontinuity and removable discontinuity. Each type can have only a countable number of discontinuous points. The set of simple discontinuity is the union of these two countable sets, and their union is also a countable set. Therefore, the set of points of simple discontinuity in \((a, b)\) is at most countable.

Step-by-step solution

Understanding the types of simple discontinuities

There are two types of simple discontinuities: 1. The function has a jump discontinuity, which means there exists an increasing function \(f\) at point \(x\) such that \(f(x-) < f(x+).\) 2. The function has a removable discontinuity, which means there exists an increasing function \(f\) at point \(x\) such that \(f(x-) \neq f(x+).\) We will deal with both types separately.

Dealing with the jump discontinuity

According to the hint provided, let \(E\) be the set on which \(f(x-) < f(x+).\) With each point \(x\) of \(E\), we associate a triple \((p, q, r)\) of rational numbers such that: (a) \(f(x-) < p < f(x+)\) (b) \(a < q < t < x\) implies \(f(t) < p\) (c) \(x < t < r < b\) implies \(f(t) > p\) Now, we need to show that there can only be a countable number of triples \((p, q, r)\). Since rational numbers are countable, the set of all rational triples is countable. We will now demonstrate that each triple \((p, q, r)\) can be associated with at most one point \(x\) in the set \(E\). Suppose there are two distinct points \(x_1\) and \(x_2\) (with \(x_1 < x_2\)) in \(E\) such that they are related to the same triple \((p, q, r)\). We see that there are two conditions that cannot be satisfied simultaneously: \(\bullet\) Condition (b) with \(x_1 < t < x_2\): We would have \(f(t) > p\) which contradicts the premise that \(f(t) < p\). \(\bullet\) Condition (c) with \(x_1 < t < x_2\): We would have \(f(t) < p\) which contradicts the premise that \(f(t) > p\). Thus, there can be at most one point in \(E\) associated with each triple \((p, q, r)\).

Dealing with the removable discontinuity

Using a similar strategy, we can define a set \(F\) on which \(f(x-) > f(x+).\) With each point \(x\) of \(F\), associate a triple \((p, q, r)\) of rational numbers such that: (a) \(f(x-) > p' > f(x+)\) (b) \(a < q' < t < x\) implies \(f(t) > p'\) (c) \(x < t < r' < b\) implies \(f(t) < p'\) Once again, since rational numbers are countable, the set of all rational triples is countable. Also, by a similar argument as mentioned earlier, we can demonstrate that each triple \((p', q', r')\) can be associated with at most one point \(x\) in the set \(F\).

Conclusion

We have shown that each type of simple discontinuity can have only a countable number of discontinuous points. Thus, the set of points at which the function \(f\) has a simple discontinuity is the union of sets \(E\) and \(F\), which are both countable sets of rational triples, and their union is also countable. Hence, the set of points of simple discontinuity in \((a, b)\) is at most countable.

Key concepts

Mathematical Analysis

Mathematical analysis is a branch of mathematics that deals with limits and related theories, such as differentiation, integration, measure, sequences, and series. The topic of our discussion involves exploring the concept of continuity and discontinuity within this field. A function is considered continuous at a point if the left-hand limit, the right-hand limit, and the function's value at that point all exist and are equal. However, when one or more of these conditions are not satisfied, we identify a discontinuity.

Most notably, mathematical analysis examines the types of discontinuities that can occur, one of which is a simple discontinuity. Our exercise hones in on proving that a particular type of discontinuity, when involving real functions, creates a set that is at most countable. This is a fundamental aspect of real analysis since it links the abstract nature of mathematical constructs like sets and functions to the more concrete properties that can be categorized and counted.

Real Function

A real function is a rule that assigns a unique real number to each element in a subset of the real numbers. Functions can display different kinds of behaviors, and one of these behaviors is discontinuity. The simple discontinuities, particularly those with jump discontinuity or removable discontinuity, are of interest in the given exercise.

For example, with a jump discontinuity, the function 'jumps' from one value to another as it approaches a certain point from the left and right, leading to a break in the graph. Conversely, a removable discontinuity often implies a 'hole' in the graph where the function value is not defined, even though the limits from both sides agree. Understanding how these discontinuities work is essential for analyzing the behaviors and characteristics of real functions within their domains.

Countable Set

In mathematics, a countable set is one that has the same size as some subset of the natural numbers. Countable sets include finite sets and infinite sets that can be put into a one-to-one correspondence with the natural numbers, such as the set of rational numbers. The idea of countability is crucial in analyzing the number of discontinuity points in our exercise.

The exercise demonstrates that every simple discontinuity point of a real function can be associated with a unique triple of rational numbers, which is a countable set. By doing so, it bridges the qualitative nature of function behaviors with the quantitative concept of countability, allowing us to conclude that such discontinuities are, in a sense, manageable and can be enumerated, a fact that has important implications for various applications in mathematical analysis.