5
$\begingroup$

Anyone ever come across a real non-textbook example of a graph with a hole in it?

In Precalc, you get into graphing rational expressions, some of which reduce to a non-rational. The cancelled factors in the denominator still identify discontinuity, yet can't result in vertical asymptotes, but holes.

Thanks!

  • 0
    What would be quite interesting to me is a function that a. has more than one removable discontinuity, and b. "naturally" shows up in applications.2011-11-30

4 Answers 4

9

Yes, this actually came up in a loan calculator I was asked to code. Given whole number $n > 0$ and real $r>0$, there is a formula for the geometric sum, $ 1 + r + r^2 + r^3 + \cdots + r^{n} = \frac{1 - r^{n+1}}{1-r}.$ This works fine when $r \neq 1$. However, it failed for $r=1$ (program crashes, and all that happy stuff), because there is a hole in the function $S(r) = \frac{1 - r^{n+1}}{1-r}$ at $r=1$. It is a removable discontinuity, but the computer didn't know that. It had to be hard-coded that: $ 1+ r + r^2 + r^3 + \cdots + r^{n} = n+1, \quad \textrm{when $r = 1$}. $

Hope this helps!

  • 0
    @JackOfAll: There's also a dependence on $n$. So for example, when $n=4$ and $r=1$, we have $1 + r +r^2 + r^3 + r^4 = 1 + 1 + 1 + 1 + 1 = 5$.2011-11-30
5

The sine cardinal function, $\dfrac{\sin\,x}{x}$. It turns up often enough in signal processing and a number of other applications.

  • 2
    Yes; that the sine cardinal has to be patched that way makes it similar to Shaun's example, no?2011-11-30
2

A car goes 60 miles in 2 hours. So 60 miles/2 hours = 30 miles per hour.

But how fast is the car going at a particular instant? It goes 0 miles in 0 hours. There you have a hole!

It is for the purpose of removing that hole that limits are introduced in calculus. Then you can talk about instantaneous rates of change (such as the speed of a car at an instant), which is the topic of differential calculus.

  • 0
    @JackOfAll : Look at the curve $y=x^3$ and ask: When $x=1$ and $y=1$, then $y$ is changing how many times as fast as $x$ is changing? That's like the instantaneous speed of a car. You can say the change in $x$ is $x-1$, and the change in $y$ is $x^2-1$, so the change in $y$ over the change in $x$ is $(x^2-1)/(x-1)$. This has a "hole" at $x=1$. If you factor the numerator and do the cancellation, you get $x+1$, and when $x=1$, this is $2$. So at that point, $y$ is changing $2$ times as fast as $x$ is changing.2011-12-01
1

I guess the derivative of the absolute value (on the reals) comes up in certain "actual" applications. It is undefined at $0$, and no way of plugging the hole makes it continuous. Which doesn't prevent one from defining arbitrarily a value of for instance $0$ at $0$, but it seems better to just leave the hole.

  • 1
    @J.M. It's closely related to the sign function, but it's unhelpful to define an arbitrary value for "the derivative of [any non-differentiable function]"2011-11-30