Examples of functions with a removable discontinuity

Question

I recently came across the following example of a function which is discontinuous at a point:

The cost of a sushi buffet is calculated by weighing your plate, where you are then charged £10 per pound. As part of a promotion, if you manage to fill your plate so that the weight is precisely 1lbs, you get your meal for free.

This is a nice way to motivate a function like $$f(x) = \begin{cases}10\,x\hfil&\text{$x\neq 1$}\\\hfil0\hfil&\text{$x=1$}\end{cases}$$ which are nice for illustrating the idea of a limit to students.

Are there any other real-world examples of functions like this, for the sake of explaining limits? Ideally one where you would actually want the limiting value, and not the assigned value, to motivate the concept further. (Apart from the obvious difference quotient.)

Answer 1

Suppose there is a carpark that charges \$5 (Singapore dollars) for the first hour and then \$1 for every subsequent half-hour, including parts of half-hours. Thus if you parked for $2.5-\epsilon$ hours you would pay \$8, but parking for $2.51$ hours would incur \$9. $$f(x)=5+\max(0,\lfloor2x-1\rfloor)$$ This function is discontinuous at infinitely many points.

For a two-sided limit, suppose there is a long, straight fence and a slit in that fence at $x=0$. The function is whether you can see beyond the fence – you cannot ($0$), even if you creep up to it (taking the limit), unless you are actually peering through the slit ($1$). $$f(x)=[x=0]$$

Answer 2

• For each math exercise with a single number as an answer, the function that gives you your grade depending in what you write is full marks if you get the number right, and $0$ if you don't. Students will probably be used to this one.

• In speedrunning of games, where you have to beat a game the fastest way possible, there are sometimes tricks called pixel perfect, where you have to do an action in a precise pixel or it won't work. Say for example the game is Super Mario Bros, and jumping in a particular pixel will let you beat the stage in 10 seconds, while missing the jump will let you beat it in 20s. The function of how much time you spend in the stage depending in what pixel you jump has a removable discontinuity.

• Consider two wi-fi routers that give you internet if you're a distance strictly less than 5 meters around them. In your house there is a long hallway of 10 meters, so you put one in each extreme of the hallway. The amount of internet you'll receive in each point will be the amount the routers give except exactly at the middle, where you'll get zero because you're 5 meters away from each one.

Some students (or anyone) could complain that these examples don't have a continuous variable (in the pixels example, or the original weight question if you take into account we can only measure a finite number of possible weights) or that the wi-fi routers don't usually behave like that, or that if you write the answer up to 20 decimals points you'll likely get a full mark in the exercise even if it's not the exact number. And I'd argue they're absolutely right, that's a consequence of living in a real world where space and almost all of our measurements of physical things can be discretized.

Answer 3

The most common example I can think of are characteristic functions. Given sets $A\subset B$, we define $\chi_A:B\to\{0,1\}$ by $$\chi_A(x)=\begin{cases}1&x\in A\\0&x\notin A\end{cases}$$

With an appropriate choice of $A$ as a discrete set in $B$, the characteristic function has a removable discontinuity at every point in $A$.

An actual real-world example is sprinting. In sprinting, we care a lot about initial acceleration, i.e., acceleration off the blocks. However, if we plot acceleration vs time in the sprint, the value of acceleration at time $0$ (the starting gun shot) is $0$, but it immediately jumps to a large value and then maintains a high value for the rest of the race. We don't really care about the fact that the acceleration at the moment the gun is shot is $0$, rather, we care about the initial push.

As a generalization of this idea, you will be able to find other examples in things like acceleration, jerk, etc, since they are not inherently continuous.