The Trouble with Continuity: three violations captured in photos -- By Kathy Jaqua

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 lim_x->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) lim_x->a f(x) exists; and

3) f(a) = lim_x->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.