Types of Discontinuities

1. An Intuitive Overview of Discontinuities

Before diving into the formal definitions, it's helpful to get a clear visual and intuitive feel for each type of discontinuity.

Let's watch a short video that provides a quick, clear introduction to these three categories.

As the video explains, we classify discontinuities into three main types. They are also grouped into two broader categories:

1. Removable Discontinuity: This is the "hole" in the graph. It's called removable because we could, in theory, "plug the hole" by defining or redefining the function's value at that single point to make it continuous. In some texts, this is also called a point discontinuity.
2. Non-Removable Discontinuities: These are more severe breaks that cannot be fixed by changing a single point. This category includes:
   • Jump Discontinuity: The function "jumps" from one value to another.
   • Infinite Discontinuity: The function shoots off towards positive or negative infinity. This happens at a vertical asymptote.

2. Formal Definitions: Connecting Discontinuities to Limits

Now, let's connect these intuitive ideas back to the formal, limit-based definition of continuity we learned in the last lesson. A function is discontinuous at $x=a$ because the condition ${\lim }_{x\to a}f(x)=f(a)$ fails. The way it fails determines the type of discontinuity.

The OpenStax resource below provides the formal definitions. We will then use a video from Khan Academy to explore the reasoning behind them.

Now, let's watch a video that explains how these definitions arise from the three conditions for continuity.

Let's summarise the key ideas from these resources:

• Removable (Point) Discontinuity:

  • What happens: The two-sided limit ${\lim }_{x\to a}f(x)$ exists (it's a finite number).
  • Why it's discontinuous: This limit is not equal to the function's value $f(a)$. This could be because $f(a)$ is undefined, or because $f(a)$ has a different value entirely.
  • It's "removable" because the function is heading towards a specific point; the hole is the only problem.

• Jump Discontinuity:

  • What happens: The one-sided limits, ${\lim }_{x\to a^{-}}f(x)$ and ${\lim }_{x\to a^{+}}f(x)$, both exist as finite numbers, but they are not equal.
  • Why it's discontinuous: Because the left-hand and right-hand limits differ, the overall two-sided limit ${\lim }_{x\to a}f(x)$ does not exist.
  • The function "jumps" because it approaches two different values from the left and the right.

• Infinite Discontinuity:

  • What happens: At least one of the one-sided limits is infinite. That is, ${\lim }_{x\to a^{-}}f(x)=\pm \infty$ or ${\lim }_{x\to a^{+}}f(x)=\pm \infty$.
  • Why it's discontinuous: The limit does not exist as a real number; the function is unbounded.
  • This occurs at a vertical asymptote.

3. How to Find and Classify Discontinuities from an Equation

We now have the theoretical tools. Let's move on to the practical skill of identifying and classifying discontinuities directly from a function's formula. This is particularly relevant for the types of functions you've worked with before, like rational and piecewise functions.

Let's return to the video from The Organic Chemistry Tutor, as it provides excellent, straightforward examples of this process.

Here's a summary of the methods for different function types:

For Rational Functions $f(x)=\frac{P(x)}{Q(x)}$

1. Find potential discontinuities: Set the denominator $Q(x)$ equal to zero and solve for $x$. These are the only points where the function might be discontinuous.
2. Factorise everything: Factorise both the numerator $P(x)$ and the denominator $Q(x)$.
3. Classify the discontinuities:
   • If a factor $(x-a)$ in the denominator is cancelled out by a matching factor in the numerator, there is a removable discontinuity (a hole) at $x=a$.
   • If a factor $(x-a)$ remains in the denominator after simplification, there is an infinite discontinuity (a vertical asymptote) at $x=a$.

For Piecewise Functions

1. Check the individual pieces: Look at the formula for each piece of the function. If any piece is a rational function, check it for discontinuities using the method above. Make sure any such discontinuity falls within the domain specified for that piece.
2. Check the "break points": The most likely places for discontinuities are the $x$-values where the function's definition changes. For a break point at $x=a$:
   • Calculate the left-hand limit: ${\lim }_{x\to a^{-}}f(x)$ (by plugging $a$ into the formula for $x<a$).
   • Calculate the right-hand limit: ${\lim }_{x\to a^{+}}f(x)$ (by plugging $a$ into the formula for $x>a$).
   • If the limits are finite but not equal, you have a jump discontinuity.
   • If the limits are equal, check the function's actual value, $f(a)$. If $f(a)$ is different from the limit or undefined, you have a removable discontinuity.

Conclusion

This lesson concludes our introduction to limits and continuity. By building on the formal definition of continuity, we have created a robust system for classifying the different ways a function can fail to be continuous.

Key Takeaways:

• Discontinuities are classified based on the behaviour of the limits around the point in question.
• A removable discontinuity (or point discontinuity) occurs when the two-sided limit exists but is not equal to the function's value. These are the "holes" in a graph.
• A jump discontinuity occurs when the left- and right-hand limits both exist as finite numbers but are not equal.
• An infinite discontinuity occurs when at least one of the one-sided limits approaches $+\infty$ or $-\infty$. This corresponds to a vertical asymptote.
• We can use algebraic techniques like factorising (factoring) for rational functions and comparing one-sided limits for piecewise functions to identify and classify these discontinuities.

You have now built a solid foundation in the concepts of limits and continuity. These ideas are the bedrock upon which calculus is built. In our next module, we will take our first major step into differential calculus by defining the derivative. You will see how limits are used to define the gradient (slope) of a tangent line to a curve, opening up a powerful new way to analyse (analyze) the rate at which functions change.