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Next week I will start teaching Calculus for the first time. I am preparing my notes, and, as pure mathematician, I cannot come up with a good real world example of the following.

Are there good examples of \begin{equation} \lim_{x \to c} f(x) \neq f(c), \end{equation} or of cases when $c$ is not in the domain of $f(x)$?

The only thing that came to my mind is the study of physics phenomena at temperature $T=0 \,\mathrm{K}$, but I am not very satisfied with it.

Any ideas are more than welcome!

Warning

The more the examples are approachable (e.g. a freshman college student), the more I will be grateful to you! In particular, I would like the examples to come from natural or social sciences. Indeed, in a first class in Calculus it is not clear the importance of indicator functions, etc..

Edit

As B. Goddard pointed out, a very subtle point in calculus is the one of removable singularities. If possible, I would love to have some example of this phenomenon. Indeed, most of the examples from physics are of functions with poles or indeterminacy in the domain.

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    $\begingroup$ This is pretty common in phase transitions, where numerous quantities such as change in energy, gibbs potential, etc. can experience discontinuities: itp.uni-frankfurt.de/~valenti/WS13-14/all_1314_chap6.pdf $\endgroup$ – Alex R. Aug 19 '16 at 21:35
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    $\begingroup$ Shock waves are also a physical phenomenon that deals with discontinuities. $\endgroup$ – nicomezi Aug 19 '16 at 21:38
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    $\begingroup$ The classic example in electromagnetism is the field of a point charge: the field is not defined at $c$. This causes all kinds of problems: to wit, delta functions. $\endgroup$ – symplectomorphic Aug 19 '16 at 21:43
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    $\begingroup$ The magnitude of the force of friction, it is discontinuous in the moment when a body starts moving, and then it's constant. $\endgroup$ – YoTengoUnLCD Aug 19 '16 at 21:46
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    $\begingroup$ Piecewise defined functions that arise in postal rates, income tax rates, taxi cab rates, etc. are examples. One example that has been in the news lately (in the U.S.) is the minimum wage increase, where in some cases it has caused some people to have LESS money because with the increase, their income crossed one or more thresholds for receiving various types of government assistance. $\endgroup$ – Dave L. Renfro Aug 19 '16 at 21:53

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If $f(t)$ is the number of humans alive at time $t$, then there is a discontinuity of $f$ at every time a human is born or dies.

The same logic holds for bank accounts, disk space, the number of molecules of caffeine in your body, and all other discrete phenomena.

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One discontinuous phenomenon in physics is the electric field above and below a surface with uniform positive charge density $\sigma$. The field above the sheet is pointing directly upwards with $E_\text{top} = +{\sigma \over 2\varepsilon_0} \hat k$ and that below being $E_\text{below} = -{\sigma \over 2\varepsilon_0} \hat k$. You could forgo the vector notation and simply use $+$ and $-$ to distinguish the two.

Reference: any college-level physics textbook, say Griffiths.

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  • $\begingroup$ Could you link a reference? I didn't know this! $\endgroup$ – YoTengoUnLCD Aug 19 '16 at 23:34
  • $\begingroup$ This is a good example. $\endgroup$ – Omar Nagib Aug 20 '16 at 15:42
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If instantaneous elastic collisions count as real world then the speed $f(t)$ of a ball traveling at constant speed $v$ and hitting a wall at $t = t_0$ is $v$ for $t \lt t_0$, $0$ at $t = t_0$, and $-v$ for $t \gt t_0$, so both lateral limits exist at $t_0$ but are different between them and different from $f(t_0)$.


[ EDIT ] As posted, $f(t)$ is considered to be the signed linear speed along the direction of movement, which is assumed to be perpendicular to the wall.

Replacing this with the magnitude of the speed $|f(t)|$ gives a function that equals $v$ for all $t \neq t_0$ and is $0$ at $t = t_0$, which is an example where $lim_{t \to t_0} |f(t)|$ exists, but is different from $|f(t_0)|$.

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  • $\begingroup$ I think OP wants examples where the limit exists but is different from the function's value. $\endgroup$ – AJY Aug 19 '16 at 22:01
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    $\begingroup$ @dxiv I like this example. Although, is speed at time $t_0$ defined? Isn't is like saying that the derivative of $|x|$ is 0 at 0? More concretely, think about the same example, but with a particle of speed $v$ hitting a particle of speed $0$. Say they collide at time $t=t_0$. Which one of the two particles is still at $t_0$? Which one has speed $v$? I feel like the speed functions are not defined at $t_0$. $\endgroup$ – Stefano Aug 19 '16 at 22:08
  • $\begingroup$ I believe you're right, @Stefano. $\endgroup$ – YoTengoUnLCD Aug 19 '16 at 22:10
  • $\begingroup$ @AJY Edited the answer to add a variation where the limit exists. $\endgroup$ – dxiv Aug 19 '16 at 22:16
  • $\begingroup$ @Stefano There is of course a lot of idealization in models like this one, which is why I started by wondering what you'd count as real world. $\endgroup$ – dxiv Aug 19 '16 at 22:22
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A simpler version of the example of @AOrtiz:
Consider a glass of water, and the function $\Delta(h)=$ density at a point $h$ cm above the surface, in gm/ml. It’s $1$ when $h<0$, $0$ (or a very small $\varepsilon$) when $h>0$, and I’m sure that you’ll agree that any reasonable definition of density at a point will return $\Delta(0)=1/2$.

I’ve never used this example in teaching, but it seems to me that it would be very thought-provoking.

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A $10$ cm length of spring steel wire with a breaking tension of 1 kg is formed into a spring of length $1$ cm.

With the spring suspended vertically, a $100$ gm weight attached to the end of the spring stretches it $1$ cm beyond its natural length. Assume Hooke's law holds until the spring is completely straightened out.

Graph the function $W(x)$ where $x$ is the number of centimeters the spring is stretched and $W(x)$ is the amount of weight in grams the spring is capable of supporting when stretched $x$ cm beyond its natural length.

What can you say about the graph of $y=W(x)$ in the vicinity of $x=9$?

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Navigation may provide interesting examples: you can fly west forever (if you don't start at one of the poles) but you can't fly north forever (from anywhere on the earth).

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Every derivative you ever saw is an example. Suppose $f'(c)$ exists. Let $g(x) = (f(x) - f(c))/(x-c).$ Then $\lim_{x\to c} g(x) = f'(c).$ But $c$ is not in the domain of $g.$ This is the primordial example.

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    $\begingroup$ But it's an uninteresting example when the derivative exists, in which case the discontinuity is removable. This is Caratheodory's characterisation of differentiability. (By the way, "primordial" doesn't meant what you think it means. You probably mean"prototypical" or "archetypical"). $\endgroup$ – Rob Arthan Aug 19 '16 at 22:46
  • $\begingroup$ @Rob Arthan: The American Heritage Dictionary lists "primary or fundamental" as the second definition of "primordial." zhw.: you have an $a$ where there should be a $c$. $\endgroup$ – symplectomorphic Aug 20 '16 at 15:40
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    $\begingroup$ @RobArthan You must be fun at parties. $\endgroup$ – zhw. Aug 20 '16 at 19:14
  • $\begingroup$ To zhw: Feed me primordial soup and I can be fun anywhere. @symplectomorphic: lexicographers make their living recording bad usages. $\endgroup$ – Rob Arthan Aug 20 '16 at 19:59
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Well, it's your chance to make a point about removable singularities. Students tend to think that the functions $x+3$ and $(x^2-9)/(x-3)$ are the same, but there is that one point where they're different. "Meh! What's one little point among so many?" (I wish) they would ask. All of differential calculus is about that one point, I (would) answer. The derivative is exactly this sort of limit. It's not really a "real world" example, but it's pretty darn concrete.

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One form of the Kronecker delta function (aka discrete delta function) is the one satisfying

$$ \delta(x) = \begin{cases} 1 & x = 0 \\ 0 & x \neq 0 \end{cases} $$

This usually crops up in discrete contexts, but it makes sense in this context too.

Another appearance of the same function is as the indicator function of the set $\{ 0 \}$. Recall that indicator functions are defined by

$$ \chi_S(x) = \begin{cases} 1 & x \in S \\ 0 & x \notin S \end{cases} $$

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While this might be controversial, I'll make the claim that no physical quantity $f(x)$ can ever usefully be thought of as having a removable singularity. By definition, a physical quantity must be physically measurable, and every measurement has an associated error. The probability (not probability density, but absolute probability) of measuring the quantity exactly at the location of the removable singularity is always zero, so we might as well always redefine the function at the singularity to be continuous. (More generally, two functions representing physical quantities are always physically equivalent if they agree except on a set of Lebesgue measure zero.) The proper mathematical formalism for naturally discrete quantities embedded into continuous space is not removable singularities but the Dirac delta function.

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