Q:

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.

A:

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.

A:

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.

A:

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 $|f(t_0)|$.

