Continuity is one of the characteristics, features, or properties of functions listed in our offical analysis list.

The definition of continuity requires more experience with limits than we have so far. However, we can begin to get a feel for contiuity by first examining functions that are not continuous.

We have several types of discontinuities and they all have a similar graphical pattern - they appear as breaks in the graph.

A break in a graph is a place in the graph where the graph is not following our expectations based on surrounding points.

The graphs we have seen so far clearly show several kinds of breaks. A quick review of graphs seen so far in this course shows there are three basic types of breaks.

Basic Types of Breaks

The function represented by the graph below is experiencing some drastic behavior around some individual domain numbers. Let’s examine some of these breaks and classify them into a few types.

• Removeable: The graph above has a very subtle break at \(t=-6\). The graph is almost one piece, except one point has been shifted vertically. This type of break gets the name removeable, because you could remove the break by just sliding that one point back in place.
• Asymptotic: The break at \(t=3\) is not subtle at all. One side is heading up to infinty and the other side is heading down to negative infinity. You can’t get a bigger break than that. Graphically, the asymptote is communicating this behavior, hence, the break is called an asymptotic break.
• Jump: The break at \(t=6\) is called a jump, because the graph makes a sudden vertical jump from one location to another location.

In all three cases, the graph experiences a sudden vertical movement. This sudden vertical movement may or may not not be predictable from the points leading up to the sudden movement.

In all three cases, the graph illustrates that there is a difference between the value of the function and the expectation of the value based on surrounding points.

We would like to communicate the difference between values and expected values.

Our language for expected values is called limits, which is a main interest of Calculus.

Discontinuities

One of the strongest purposes of mathematics is simply description and communication. We like mathematics to be explicit and exact, which requires the support of lots of specific language and notation. We call this rigor. As we investigate breaks in the graphs of functions, we want language that is explicit, so that we can clearly communicate about the underlying function behavior.

It is not enough to a basic idea. We want to be exact and we want to express our thoughts on exactness so that other people can understand what we see.

With this in mind, let’s compare the following two graphs.

These two graphs (and the underlying functions) are almost identical. They have the same breaks, almost.

The difference is the graph on the left has a point for \(x=6\) and no point on the vertical asymptote. The right graph has reversed this.

The difference is the underlying function for the graph on the left includes \(6\) in its domain and not \(3\). The underlying function for the graph on the right reverses this. It includes \(3\) and not \(6\) in its domain. In the world of functions, this is significant. We want our language to note the differences.

Therefore, we are going to adopt some language to separate these ideas.

\(\blacktriangleright \) Discontinuities are numbers in the domain of the function and singularities are not in the domain.

Of course, all of this gets more complicated as we encounter more functions. But this is a good start.

We will soon algebraically define discontiuities to match these ideas.

Discontinuities

Let \(K(y)\) be a function. The graph of \(z = K(y)\) is displayed below.

Select all of the discontiuities of \(K\).

\(-8\) \(-3\) \(-1\) \(3\) \(4\) \(8\) None of the above

\(K(t)\) has one discontinuity. The idea of a discontinuity is that the value of the function changes abruptly at a domain number. The graph jumps vertically.

There is a vertical jump at \(4\). However, \(4\) is not in the domain of \(K\). So, \(4\) is a singularity, not a discontinuity.

There is a vertical jump at \(-3\). And, \(-3\) is in the domain of \(K\). So, \(-3\) is a discontinuity.

Discontinuities are domain numbers.

Discontinuities

Let \(h(t)\) be a function. The graph of \(y= h(t)\) is displayed below.

Select all of the discontiuities of \(h\).

\(-8\) \(-3\) \(-1\) \(3\) \(4\) \(8\) None of the above

\(h(t)\) has no discontinuities. The idea of a discontinuity is that the value of the function changes abruptly at a domain number. The graph jumps vertically.

In the graph above, there is horizontal space between the graphical pieces.

The intervals in the domain of \(h\) are separated by unused intervals of real numbers. There is no abrupt vertical jump. There is just an end and then a new beginning further along.

Discontinuities

Let \(K(m)\) be a function. The graph of \(y= K(m)\) is displayed below.

\(K(m)\) has two discontinuities, which occur at the endpoints of two maximal intervals in the domain.

Select all of the discontiuities of \(K\).

\(-8\) \(-3\) \(-1\) \(3\) \(4\) \(8\) None of the above

\(\blacktriangleright \) We want to say that there are ALWAYS domain numbers on one or both sides of the discontinuity no matter how close you get to the discontinuity.

How are we going to talk about “no matter how close you get”, algebraically?

The last condition says that no matter how close to \(c\) you look in the domain, there is always another domain number where the function value is wildly different from \(f(c)\).

No matter how close you look at \(c\), the jump is still there.

We need algebraic language for “no matter how close you get in the domain” and ‘no matter how close you get in the codomain”.

We will slowly build this language.

But, for sure, if \(c\) is a discontinuity of \(f\), then we say that \(f\) is discontinuous at \(c\).

Discontinuity

Let \(P(w)\) be a function. The graph of \(y = P(w)\) is displayed below.

Select all of the discontiuities of \(P\).

\(-8\) \(-4\) \(-2\) \(-1\) \(3\) \(4\) \(8\) None of the above

Jump Discontinuity

Let \(g(k)\) be a function. The graph of \(y= g(k)\) is displayed below.

• \(4\) is a discontinuity of \(g(k)\).

  First, \(4\) is in the domain of \(g\).

  For the second condition, let \(d = 1\). Any small open interval of real numbers containing \(c = 4\) must also contain an interval of the form \((4-\epsilon , 4+\epsilon )\), where \(0 < \epsilon < 0.1\). Such an interval would ALWAYS contain the domain number \(e = 4-\frac {\epsilon }{2}\).

  \[ \left | g(4) - g\left (4-\frac {\epsilon }{2}\right ) \right | > 1 = d \]

  Therefore, \(4\) is a discontinuity of \(g\).

• \(-2\) is not a discontinuity of \(g\), because we can find an open interval of real numbers that does not contain a domain number different from \(-2\). No matter what \(d\) you select, the open interval \((-2.01, -1.99)\) does not contain a different domain number, \(e \ne -2\), such that \(|g(-2) - g(e)| > d\). Therefore, the second condition for discontinuities does not hold. \(-2\) is not a discontinuity.
  

• \(8\) is a discontinuity of \(g(k)\). First, \(8\) is in the domain of \(g\). For the second condition, let \(d = 1\). Any small open interval of real numbers containing \(c = 8\) must also contain an interval of the form \((8-\epsilon , 8+\epsilon )\), where \(0 < \epsilon < 0.1\). Such an interval would ALWAYS contain the domain number \(e = 8-\frac {\epsilon }{2}\).

  \[ \left | g(8) - g\left (8-\frac {\epsilon }{2}\right ) \right | > 1 = d \]

  Therefore, \(8\) is a discontinuity of \(g\).

This is the type of exactness we want in our discussions. This is rigor. We want to talk algebraically about “always”, “close enough”, and “small enough”. Fluency in the notational language will take some time. But we have some time until Calculus.

Removeable Discontinuity

Let \(T(y)\) be a function. The graph of \(z = T(y)\) is displayed below.

• \(7\) is not a discontinuity of \(T\), because \(7\) is not a member of the domain of \(T\). (\(7\) is a singularity.)

• \(-6\) is a discontinuity of \(T(y)\). \(-6\) is in the domain of \(T\). For the second condition, let \(d = 0.5\). Any small open interval of real numbers containing \(c = -6\) must also contain an interval of the form \((-6-\epsilon , -6+\epsilon )\), where \(0 < \epsilon < 0.1\). Such an interval would ALWAYS contain the domain number \(e = -6+\frac {\epsilon }{2}\).

  \[ \left | T(-6) - T\left (-6+\frac {\epsilon }{2}\right ) \right | > 1 > 0.5 = d \]

  Therefore, \(-6\) is a discontinuity of \(T\).

• \(4\) is a discontiuity of \(T(y)\). \(4\) is in the domain of \(T\). For the second condition, let \(d = 0.5\). Any open interval of real numbers containing \(c = 4\) must also contain an interval of the form \((4-\epsilon , 4+\epsilon )\), where \(0 < \epsilon < 0.1\). Such an interval would ALWAYS contain the domain number \(e = 4-\frac {\epsilon }{2}\).

  \[ \left | T(4) - T\left (4-\frac {\epsilon }{2}\right ) \right | > 1 > 0.5 = d \]

  Therefore, \(4\) is a discontinuity of \(T\).

No matter how close you get to \(-6\) in the domain, there is ALWAYS a jump of more than \(1\) in the value of the function.

Asymptotic Discontinuity

Let \(f(x)\) be a function. The graph of \(y = f(x)\) is displayed below.

• \(3\) is not a discontinuity of \(f\), because \(3\) is not a member of the domain of \(f\). (\(3\) is a singularity.)

• \(-5\) is a discontinuity of \(f(x)\). For the second condition, let \(d = 0.5\). Any open interval of real numbers containing \(c = -5\) must also contain an interval of the form \((-5-\epsilon , -5+\epsilon )\), where \(0 < \epsilon < 0.000001\). Such an interval would ALWAYS contain the domain number \(e = -5-\frac {\epsilon }{2}\).

  \[ \left | f(-5) - f\left (-5-\frac {\epsilon }{2}\right ) \right | > 0.5 = d \]

  Therefore, \(-5\) is a discontinuity of \(f\).

Discontiuity and Singularity

Let \(h(v)\) be a function. The graph of \(y = h(v)\) is displayed below.

\(4\) is a discontinuitysingularity of \(h\).

\(5\) is a discontinuitysingularity of \(h\).

Is \(-1\) a discontiuity, sigualrity, or neither of \(h\).

discontinuity singularity neither

As you can see, this is easy to see visually, but difficult to describe algebraically. It will take some time to describe discontinuities and singularities with algebra.

Core and Elementary Functions

All of the core and elementary functions are continuous, except for GIF.

They are all continuous on their domains.

This makes studying discontinunities difficult using core and elementary functions. The only tool we have to help us study discontinuities is piecewise defined functions.

We use piecewise defined functions a lot!