Sometimes in mathematics we provide a formal definition of a
concept, but a newcomer to that idea has difficulty understanding the nuances
of the definition. Discontinuity of a
function at a point is one of those ideas.
Formally, a function is continuous at a point if the limit of the
function at that point is equal to the value of the function at that point or limx->a f(x) = f(a). Hidden in this definition are three
conditions that could provide a way for a function to be discontinuous.
1) f(x) exists at x=a;
2) limx->a f(x) exists; and
3) f(a) = limx->a f(x)
Even these
conditions can overwhelm a lot of students, so an informal way of thinking about
discontinuity is given based on graphing: “If you can’t draw the graph of the
function without lifting your pencil, then the function is discontinuous where
you have to lift your pencil.” That
still misses the nuances of the definition, so we continue by describing points
of discontinuity as “holes,” “jumps” and “asymptotes,” to visualize what
happens at the problem points where you lift your pencil. “Holes” violate condition 1 because a hole in
the graph means that f(x) is undefined at that x value. “Jumps” violate condition 2 because if the
graph doesn’t match on both sides of a given x value, then you must make an
instantaneous leap in function values at a point, and thus a limit at the point
does not exist. “Asymptotes” also
violate condition 2 because the function gets infinitely large or small near a
specific x value and so the limit does not exist. The final condition accounts for when both of
the first conditions hold, but they just don’t have the same value.
On a recent
math selfies assignment, students
submitted photos that show violations of each of these conditions for
continuity. Their explanations clearly
describe which condition is violated, and use the language of “holes” and
“jumps” to explain those violations. I
hope you enjoy this view into students’ understanding of what makes a function
discontinuous.
Condition 1
violation: f(x) does not exist or a hole
in the graph
 |
| figure 1: A hole in the power line |
“These are power lines and these power lines are an example of
discontinuity. … In the case of the power lines in the example [they] are
continuous until they reach the power poles. The [power] lines in the picture
represent the plotted points on a line and as each line reaches a power pole it
represents a place of discontinuity or a hole in the ‘graph’.”
The author
of this math selfie has represented
the idea that at specific x values
(in this case the poles) the function values (in this case the power lines) are
missing as the power lines are cut to reconnect with the equipment on each
pole. The power lines and power poles
can be visualized as part of a graph where the edges of the photo are the two
axes. Even though the student sees each
section as a line (in the graphical sense), it is true that each section is a curve. That means that a piece-wise function
could be written to model the visualized graph.
Condition 2 violation: lim xàc
f(x) does not exist or a jump in the graph
In this math selfie,
the author has concentrated on the treads of the steps to see the jump from
each tread to the next as one moves down the steps. In this case, there is not the sense of the
outline serving as a graph based on the edges of the photo, but there is still
the notion of input and output. Here the
sense of “jumping” is the change in height for each step. The author also notes the problem that there
could be two different heights for the specific spots on the steps and even the
possibility of an infinite number of heights in the vertical rise of each step.
 |
| figure 2: Jumping down the steps |
“These
stairs are an example of discontinuity because there are several spots where
the steps go down at a 90 degree angle as if the x values at these spots has
multiple y values. Discontinuity in a
graph occurs when there is a jump in the line…”
Condition 3
violation: the limit and the value of
the function don’t agree
In this photo,
the author moves away from the idea of graphs completely. Now we are looking at a function as a
collection of inputs and outputs. The book’s
missing pages (pages 57 and 58) mean that we will have to look closely at what
happens here. If we were simply turning
pages of the book starting from before the illustrated pages, we would see page
53, then 54, then 55, then 56, and we could infer that pages 57 and 58 should
be next. If we turned pages in reverse
from past the illustrated pages, we would see page 62, then 61, then 60, then
59, and again we would infer that pages 57 and 58 should be next. So the idea that the limit of our page
turning from before or after should lead to pages 57 and 58 being visible. But clearly what we have visible are pages 56
and 59. So the limit, pages 57 and 58, is not the same as the output, pages 56
and 59.
 |
| figure 3: Missing pages |
“For a function to be
continuous it must have a limit, the function must exist, and the limit and the
output must be the same. The first picture is an example of discontinuity
because page 57 and 58 do not exist; therefore, the output and the limit are
not the same.”
I found
this photo to be one of the most interesting ones submitted because it got away
from graphs or calculations to show this lack of equality and still captured
the sense that the predicted outcome, or limit, did not match the observed
outcome, or function value. The
creativity of this interpretation showed a deeper level of understanding of
this condition of continuity than just the idea of “drawing a graph without
lifting the pencil.” These authors
provided good math selfies that
clearly show their own visualizations of a complex mathematical idea.
