In electrical engineering, I have encountered two situations, one that I know how to understand mathematically, the other which I do not.
The first, which I understand, is the voltage drop over a resistor in a circuit with just a battery (constant voltage) and that resistor.
We are told that no matter how big the resistor is, the voltage drop will always be the opposite of the battery, but if the resistor has no resistance (a.k.a. is just a wire) then there is no voltage drop over the resistor.
This can be written as
$$ f(x) = \left\{ \begin{array}{ll} -b & \quad x \gt 0 \\ 0 & \quad x = 0 \end{array} \right. $$
where x is the voltage drop over the resistor, and b is the voltage drop over the battery
So, $$\lim_{x\to0^{+}} = -b$$ and the function is merely discontinuous at 0. That makes sense.
What doesn't make sense is the same situation with a constant current source.
Here, we are told that when the resistor has zero resistance, the current running through the wire stays at the constant value of the source. As we increase the resistance, the current stays at the same constant value.
However, if the resistance is infinitely large (a.k.a. the resistor is pulled out), then the current drops to 0.
Is this representable as a mathematical function? It seems as though checking if x is infinity is weird. For example, is the following a mathematical function?
$$ g(x) = \left\{ \begin{array}{ll} c & \quad 0 \le x < \infty \\ 0 & \quad x = \infty \end{array} \right. $$
What mathematical representations are available for a concept like this? (If the above or something similar does not suffice.)