Examples of limits in nature with $\lim_{x \to c}f(x) \neq f(c)$

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.

Solution 1:

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.

Solution 2:

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.

Solution 3:

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 $f(t_0)$.

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 $|f(t_0)|$.

Solution 4:

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.

Solution 5:

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 $x=9$?