# Continuity

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- Understand continuity at a point or over an interval conceptually and visually based on graphs
- Define continuity of a function at a point using limits
- Define continuity of a function over an interval using limits
- Classify types of discontinuities, either based on graphs or based on function definitions

We've talked in the past in Algebra Two and Pre-Calculus about a loose definition of what it means for a function to be continuous. Now we'll talk about a rigorous definition, using limits.

Practice problems and worksheet coming soon!

## It's Good to Be Well-Connected

Continuity is a concept that could be studied and understood in Algebra, but is more acutely defined in here in Calculus, through the use of limits. So what exactly is Continuity all about? In short, it's a measure of whether a function is connected with itself. We'll see that this is easier to define visually / graphically than it is to define algebraically, though we will need to be experts on both for typical exam questions.Let's digest this concept first with a loose definition and a focus on the graphical approach, before moving to the more rigorous and tedious algebra based tasks.## A Practical Definition

Functions are continuous along the parts of their graph that you can draw without picking up your pencil, and discontinuous at the place(s) where you had to pick up your pencil. This idea is the loose definition of continuity.Here is a graph of a function that has a point of discontinuity, followed by a graph that is continuous everywhere on the function's domain:Note that $f(x)$ has a gap when $x=3$, and how it would not be possible to draw this graph without picking up your pencil. This is not a rigorous definition of continuity, but it is true that $f(x)$ is not continuous at $x=3$. It is also true that $f(x)$ is continuous everywhere else that we can tell. This leads is to our first important point about continuity:Continuity is not a yes or no question - i.e. a function is not either continuous or discontinuous identically. A function is continuous on the intervals that are connected, and discontinuous at the points or intervals on the graph that we would have had to pick up our pencil when drawing.## What's the Real Question?

Let's take a look at a few more graphs to better understand what continuity and discontinuity each look like.This graph has two discontinuities - at $x=-1$ there is a hole in the graph, and at $x=2$ there is a jump in the graph. The question we're often going to need to answer is - "where is the graph continuous"? Since most parts of most graphs you will see are continuous, this question is best answered by taking the perspective that the function is continuous everywhere except for the places we identify that are discontinuous. So the graph above looks to be continuous everywhere except for $x=-1$ and $x=2$. If we are asked to present our answer in interval notation, then we would say $x$ is continuous on the interval $(-\infty, -1) \,\, \bigcup \,\, (-1, 2) \,\, \bigcup \,\, [2, \infty)$.Here's another one:While much of this graph is smooth and connected, there are a few places that stand out as "break points" or as places where, when drawing the function, we had to lift our pencil up off the page. The function is discontinuous between $x=-1$ and $x=2$, not just at the specific locations of $x=-1$ and $x=2$, because the function is not defined at all on this interval. Therefore, when asked, we would say that this function is continuous on the interval $(-\infty, -1) \,\, \bigcup \,\, (2,5) \,\, \bigcup \,\, (5, \infty)$.## The Four Flavors of Discontinuity

Hopefully with these few examples, it is becoming clear how to visually determine the places or intervals on a function that are continuous, and those that are discontinuous. Before we look at the limit definition of how to determine continuity, let's specify the four categories of discontinuities that we will encounter.Jump DiscontinuitiesThe first type of discontinuity is called a jump discontinuity. This is the classic "gap" situation where, at the point of discontinuity, each side of the function is approaching different values.Note that in terms of continuity at that point of separation, it doesn't matter which side has the solid dot where the function is actually defined, nor does it matter if neither is solid - a discontinuity will exist at this point regardless. This type of discontinuity is typically exclusive to piecewise defined functions.Removable DiscontinuitiesThe next type of discontinuity is called a removable discontinuity. Both of the following similar functions exhibit this type of discontinuity:The name comes from the idea that the function would be otherwise continuous in the vicinity of the point, were we to "plug the hole". In other words, each side of the function approaches the same value, but the function is not continuous because for whatever reason, the function value at that point is defined to be a different value, or not defined at that point at all (the first and second graphs above, respectively). The function is discontinuous either way, since the function has a puncture in it. Typically this happens naturally in rational functions that have a common factor in both the numerator and denominator (see the lesson on limits with zero in the denominator »).Infinite DiscontinuitiesUp third, we see one of the more common situations that occur in functions that we work with often.An infinite discontinuity occurs any time a vertical asymptote appears. This discontinuity category applies regardless of whether the graph approaches the same or different directions on each side of the asymptote. Both asymptotes in the following figure are infinite discontinuities.Typically, this type of behavior can appear naturally in rational functions that have a division by zero domain error. However, along with many of the stranger graphs in math, it can be a product of a piecewise defined function. In fact, it's possible to define a function piecewise such that one side of an asymptote has infinite behavior, while the other side has finite behavior. Here's an example of that usual case:Infinite Oscillation DiscontinuitiesLast but not least, we have one of the least frequent occurrences in functions we work with, as this issue is specific to trig functions. An infinitely oscillating discontinuity occurs when a periodic function oscillates with increasing frequency that becomes infinitely frequent at a specific instant. An example would be $f(x)=\sin(1/x)$ at $x=0$:Of all the discontinuity types, this one is easily the most unique, but it is also infrequent. Some teachers don't even cover it in fact, just to save time and energy.Note: while it is important and more palatable to digest the differences among these four classifications visually, we will also need to look at how to tell these four categories apart without graphs, using only the algebraic limit definition that we will turn to next. First we'll acquaint ourselves with the limit defintion approach, and then we will quickly revisit each of these four situations and see how to tell them apart with the limit method.## The Proper Approach

While most students find it easier to describe continuity with a visual approach, we must ultimately define it symbolically, using limits.Type | Description |

Jump | Each of the one-sided limits are finite but disagree |

Removable | The two one-sided limits agree but the function value does not agree |

Infinite | One or both of the one-sided limits is either $+\infty$ or $-\infty$ |

Oscillating | Typically examining some form of $\sin(\theta)$ or $\cos(\theta)$ at points where $\theta=1/0^+$ or $\theta=1/0^-$ |

## One-Sided Continuity

Just like limits can be one-sided, so too can continuity. Knowing and understanding this nuance is less important than understanding everything we've mentioned up to this point, and many teachers will not even discuss one-sided continuity, so if you don't need to know about it, jump down to the next heading. If you do, read on!Jump discontinuities may be considered continuous on one side by a very similar definition to the general limit definition of continuity, but using one-sided limits:## Well-Behaved Functions

By now, you may have noticed that many of the continuity problems we are describing are specific to specialized functions, like piecewise, rational, or trigonometric functions. This is no coincidence. Many common functions, such as polynomials, have no instances of discontinuity at any point. In fact, several of the major function families have continuity behavior that can be summarized generally.- Linear and Constant Functions
- Polynomial Functions
- Rational Functions
- Exponential Functions
- Logarithmic Functions
- Trigonometric Functions

## Mr. Math Makes It Mean

The last consideration about continuity that we need to make is the case where some function $h(x)$ is defined as some combination of two (or possibly more) other functions, say $f(x)$ and $g(x)$. This could happen by adding, subtracting, multiplying, or dividing $f$ and $g$, or it could be a result of composition, e.g. $h(x) = f(g(x))$. In any case, there is nothing too surprising happening here.## Put It To The Test

Identify Continuity - Graph BasedAmong the warm-up questions we should expect are questions asking us to identify which continuity type is which, based on the graph:Instructions for Examples 1-3: Determine the intervals on which the function is continuous, and determine the category of discontinuity that occurs anywhere that the function is discontinuous.- Understand what continuity is visually, conceptually, and symbolically
- Be able to classify discontinuity types based on graphs
- Be able to classify discontinuity types without graphs, by analyzing with limits
- Concisely describe the places or intervals on which a function is continuous
- If required, be able to identify one-sided continuity

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