Question

In Exercises \(1-10,\) find the points of continuity and the points of discontinuity of the function. Identify each type of discontinuity. $$y=|x| / x$$

Short answer

The function \(y=\frac{|x|}{x}\) is discontinuous at x = 0 and the discontinuity is of type 'jump discontinuity'. Except at x = 0, the function is continuous for all other real numbers.

Step-by-step solution

Identify the Function

The function given here is \(y=\frac{|x|}{x}\). It is ratio of absolute value of x and x. From this you can see that the function is undefined when \(x = 0\).

Find the Limits

Next, find the limits as x approaches 0 from both the left and the right sides to verify the continuity. Hence, we find \(\lim_{{x \to 0^-}} \frac{|x|}{x} \) and \(\lim_{{x \to 0^+}} \frac{|x|}{x}\). For \(x < 0\), \(|x| = -x\) and for \(x > 0\), \(|x| = x\). Thus, \(\lim_{{x \to 0^-}} \frac{|x|}{x} = \lim_{{x \to 0^-}} \frac{-x}{x} = -1\) and \(\lim_{{x \to 0^+}} \frac{|x|}{x} = \lim_{{x \to 0^+}} \frac{x}{x} = 1\). Thus, right and left limits at x = 0 are not equal, so the function is not continuous at x = 0.

Identify the Type of Discontinuity

Lastly, identify the type of discontinuity that occurs at x = 0. Since the left and right limits exist but are not equal, we call this a jump discontinuity.

Key concepts

Discontinuity in Functions

Understanding the concept of discontinuity in functions is crucial for mathematics students. A function is said to be discontinuous at a certain point when it is not continuous there. In simple terms, if you were to draw the function without lifting your pen off the paper, reaching a discontinuity would be a point where you have to lift your pen.

There are different types of discontinuities, with some of the most common being point discontinuity, jump discontinuity, and infinite discontinuity. Point discontinuity occurs when a function is undefined at a point, or its limit does not equal the function's value. Jump discontinuity, as encountered in our exercise with the function \(y=|x| / x\), occurs when there are different limits from either side of the point. Infinite discontinuity refers to when the function approaches infinity as it nears a certain point. By exploring these types, students gain a deeper appreciation for the complex behavior of functions.

Limits and Continuity

The topics of limits and continuity are intertwined in mathematical analysis. A limit is an expected value that a function approaches as the input approaches some value. For a function to be continuous at a point, three conditions must be met: the function must be defined at the point, the limit of the function as it approaches the point must exist, and finally, the limit of the function must equal its value at that point.

In the case of the function \(y=|x| / x\), we check the limits as \(x\) approaches 0 from the negative and positive sides to assess continuity. Since the limits \(\lim_{{x \to 0^-}} \frac{|x|}{x}\) and \(\lim_{{x \to 0^+}} \frac{|x|}{x}\) yield different results, we conclude that the function \(y=|x| / x\) is not continuous at \(x=0\).

Absolute Value Functions

Absolute value functions are a staple of algebra and precalculus, represented by \(y = |x|\). The absolute value symbol denotes the distance of \(x\) from 0 on the number line, always as a non-negative value. The graph of an absolute value function is a V-shaped curve that points upwards, intersecting the \(y\)-axis.

The function in our exercise, \(y=|x| / x\), involves dividing the absolute value of \(x\) by \(x\) itself, which introduces an interesting behavior. For \(x > 0\), the function simplifies to \(y=1\), and for \(x < 0\), it simplifies to \(y=-1\). This illustrates perfectly how an absolute value function can interact with algebraic operations to produce different results on either side of a discontinuity.

Jump Discontinuity

Jump discontinuity is a specific type of discontinuity that is particularly easy to visualize. It occurs when the left-hand limit and the right-hand limit of a function at a particular point exist but are not equal to each other. In other words, the function makes a 'jump' from one value to another.

The function \(y=|x| / x\) displays a jump discontinuity at \(x=0\), as the limits from the left and the right, \(\lim_{{x \to 0^-}} \frac{|x|}{x} = -1\) and \(\lim_{{x \to 0^+}} \frac{|x|}{x} = 1\), do not coincide. This results in a gap in the graph, resembling a jump, and hence the name. Recognizing jump discontinuity helps students to analyze functions and understand their graphical representations more comprehensively.