Discontinuity

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sudden breaks

The graphs we have seen so far clearly show all kinds 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.

Basic Types of Breaks

A quick review of graphs seen so far in this course shows there are three basic types of breaks.

Removeable: The graph above has a very subtle break at $t=-6$. The graph is almost one piece, except one point has been shifted up. This type of break gets the name removeable, because you could remove the break by just sliding that 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 not be predictable from the points leading up to the sudden movement.

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.

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 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.

Language

While graphs have breaks, functions will have discontinuities and singularities.

If the domain of a function includes the number where the graph is experiencing a break, then we will say the function has a discontinuity at this domain number.
If the domain of a function does not include the number where the graph is experiencing a break, then we will say the function has a singularity at this non-domain number.

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.

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, $4$ is in the domain of $K$. So, $4$ 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 $ Endpoints make an exact definition of discontinuity messy, because we want to say there are domain numbers on one or both sides of the discontinuity and these domain numbers have no space between them. There is no open interval between domain numbers.

Discontinuity

A real number, $c$, is said to be a discontinuity of the function $f$, provided

$c$ is a member of the domain of $f$, and
there is a distance, $d > 0$, such that EVERY open interval of real numbers containing $c$ also contains a different DOMAIN number, $e \ne c$, with $ |f(c) - f(e)| > d $.

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.

No matter how close you look at $c$, the jump is still there.

The second condition in the definition is our algebraic way of saying, “no matter how close you look”.

Language

Not Discontinuous and contiuous do not mean the same thing.

Not Discontinuous at $c$ means there is not a discontinuity at $c$. There can be different reasons for this.

If $c$ is not in the domain of the function, then $c$ is automatically not a discontiuity.

$c$ could turn out to be a singularity.
$c$ could turn out to just not be in the domain, but not a singularity.

Or, $c$ might be in the domain and the function might be continuous at $c$.

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 $f$.

But, But, But, ...

The example above illustrates that the difference between a real number being a discontinuity or not can be very thin.

This is why we also have the singularity category.

A Singularity

A real number, $c$, is said to be a singularity of the function $f$, if $c$ is not a member of the domain of $f$, however, $c$ is surrounded by the domain or is an edge of the domain, and the function has extreme/weird/unexpected behavior there. ($c$ is not in the domain, but it wouldn’t be a singleton if it were included in the domain.)

$\blacktriangleright $ $c$ is not in the domain, however,

EVERY open interval containing $c$ does contain a domain number and
the function is behaving extreme/weird/unexpected near $c$.

It turns our “weird” includes a lot of behavior. It will take a while to state a rigorous definition of singularity. In the mean time, we will keep noting weird behavior and building our feeling of weird.

We will extend and sharpen our ideas and definition of singularity in Calculus.

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

Continuity

Let $f$ be a function.
If $c$ is in the domain of $f$ and $c$ is not a discontinuity, then $f$ is said to be contiuous at $c$.

If $S$ is a subset of the domain and $f$ is continuous at every number in $S$, then $f$ is said to be continuous on the subset.

A function that is continuous on its whole domain is just said be a continuous function.

A function is not continuous at a singularity, because the singularity is not in the domain. However, the singularity is not a discontinuity, because the singularity is not in the domain.

So, it is possible for a function to be noncontinuous and yet not have discontinuity.

Continuity

Continuous means not discontinuous in the domain.

Discontinuous means there was a fixed distance in the range that separated function values no matter how close the domain numbers were.

Continuity means this doesn’t happen.

Let $f$ be a function and $c$ a number in the domain.

$f$ is continuous at $c$ if

For every range distance $\epsilon > 0 $, there is a corresponding domain distance $\delta > 0$ such that

for all domain numbers $x$ inside $(c - \delta , c + \delta )$ we have $| f(x) - f(c) | < \epsilon $.

You can get close enough to $c$ in the domain so that all of the function values are close to $f(c)$...as close as you want...any closeness.

Continuity: Close in the domain means close in the range.

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

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More Examples of Visual Features

2025-01-07 03:41:42