# How can you represent a function which is discontinuous at infinity, but continuous everywhere else?

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.)

• To start with (assuming you are working in $\mathbb{R}$), $\infty$ is not a number, so for that reason alone your last function doesn't make sense. From what you write now, it seems that your function $f$ is simply the constant function $f=c$, and then $\lim_{x\to\infty}f=c$. – B. Pasternak Nov 9 '17 at 6:41
• In the complex plane you can construct such, but not in real I think. You need to construct the function which will be discontinuous on the Riemann sphere pole(s). – Gevorg Hmayakyan Nov 9 '17 at 6:47
• I think you can consider the extended real line, which is $\mathbb{R}\sqcup\{\pm\infty\}$, and then use a step function (but I don't know how it precisely works with continuity and all that if you do that). Also, what @GevorgHmayakyan says (but I am not sure if that's what you are looking for). – B. Pasternak Nov 9 '17 at 6:47
• As aside, neither limiting situation makes sense if you want to be pedantic. A non zero voltage source connected across a zero resistor cannot exist just as $0=1$ makes no sense. The dual situation exists for a disconnected current source. Your definition of $g$ is perfectly fine (on the extended line) as a mathematical object, but it is not clear what utility it has. – copper.hat Nov 9 '17 at 7:00
• @ProQ you can use math to model the real world. But a model is never the real thing. For example, for all we know, continuum is just a fancy invention of our mind. – user251257 Nov 9 '17 at 9:31