Question

Continuity of Combinations of Functions If the functions \(f\) and \(g\) are continuous for all real \(x,\) is \(f+g\) always continuous for all real \(x ?\) Is \(f / g\) always continuous for all real \(x ?\) If either is not continuous, give an example to verify your conclusion.

Short answer

The sum of continuous functions \(f\) and \(g\) will always be continuous. The ratio \(f/g\) will generally not be continuous for all real \(x\), as there may be points where \(g(x)\) equals zero, resulting in an undefined ratio. An example is the functions \(f(x) = x\) and \(g(x) = x^2\), where the ratio \(f/g = 1/x\) is not defined at \(x = 0\).

Step-by-step solution

Continuity of Function Addition

To begin, consider the sum of the functions, \(f + g\). The sum of two continuous functions is continuous at a point if the functions being summed are continuous at that point. This is a property of continuous functions. Thus, \(f + g\) is always continuous for all real \(x\) if \(f\) and \(g\) are continuous for all real \(x\).

Continuity of Function Division

Next, consider the ratio of the functions, \(f/g\). A function division is continuous at a point if the denominator function at that point is not zero and both the numerator and the denominator are continuous. If \(g(x)\) equals zero for any real number \(x\), the division will not be defined, and the continuity will be disrupted at that point.

Example of Discontinuous Function Division

Consider \(f(x) = x\) and \(g(x) = x^2\). Both functions are continuous for all \(x\). However, the ratio \(f/g = x / x^2 = 1/x\) is not defined at \(x = 0\). Therefore, although \(f\) and \(g\) are continuous for all real \(x\), the ratio \(f/g\) is not continuous for all real \(x\).

Key concepts

Function Addition

Understanding the addition of functions is crucial for exploring the behavior of combined functions. When adding two functions, denoted as \( f + g \), students need to recognize that the resultant function also has a value at each point that is the sum of the corresponding values of \( f \) and \( g \) at that point. If both \( f \) and \( g \) are continuous across all real numbers, their sum \( f + g \) inherits this continuity. Therefore, function addition does not introduce discontinuities; it preserves the continuous nature of the individual functions involved, as evidenced by the sum being well-defined for every real number in the domain. This seamless behavior is important for creating complex functions that maintain predictable behaviors across their domains.

To elaborate, consider \( f(x) \) and \( g(x) \) as continuous functions. At any given point \( x_0 \), if we can find the function values \( f(x_0) \) and \( g(x_0) \) without any interruption in their respective graphs—meaning without any jumps, breaks, or holes—their sum at \( x_0 \) is simply \( f(x_0) + g(x_0) \) and is equally continuous at that point. This results in the combined function \( f + g \) forming a continuous curve when graphed on a coordinate plane.

Function Division

Division of functions, expressed as \( f / g \), is a more delicate operation compared to addition. To be continuous, the quotient function requires not only that both \( f \) and \( g \) are continuous but also that the denominator function \( g \) does not take a value of zero at any point in the domain under consideration. The reason for this is that division by zero is undefined in mathematics. Therefore, the continuity of the quotient function \( f / g \) is conditional upon the non-zero behavior of \( g \).

The critical insight for students here is that a zero in the denominator is a 'deal-breaker' for continuity. Even if the numerator \( f \) and the denominator \( g \) are smooth and continuous, a single point where \( g(x) = 0 \) can disrupt the entire flow of the quotient function, creating a vertical asymptote—a line that the graph of the function approaches but never touches or crosses. This caveat must be kept in mind when analyzing the continuity of functions formed through division.

Discontinuous Functions

Discontinuous functions can be somewhat counterintuitive. They offer valuable lessons on the limits of function operations. As mentioned, a divide-by-zero scenario is a classic example of why division of continuous functions might yield a discontinuous function. Discontinuities can be of various types, such as point discontinuities, where there is a hole in the graph, and jump discontinuities, where there is a sudden leap in function values. The existence of a discontinuity means the function does not have a well-defined output for every input within a certain interval or at specific points.

When dealing with discontinuous functions, students learn that not all mathematical operations will lead to 'neat' results. Recognizing and handling such irregularities forms a fundamental part of advanced calculus and furthers the understanding that continuity is a property that is easily disrupted and sometimes hard to restore.

Real Numbers

Real numbers encompass both rational and irrational numbers, forming a continuous line on the number scale, known as the real number line. They are the most fundamental building blocks for defining the domain and range of functions in calculus. When we talk about functions being continuous for all real \( x \), we imply that for every real number within the domain, there exists a corresponding real number as the output, without any gaps or jumps.

For students, grasping the concept of real numbers is vital because it offers a concrete base where other abstract mathematical concepts can be grounded. Any function defined on real numbers that provides a real output without interruption is considered to be continually mapping the domain to the range. The elegance of real numbers lies in their ability to represent a vast variety of values, including those that come out of function operations like addition and division, as long as they are defined and free of discontinuities.