(Honesty and mathematical integrity - given the number of votes on this "answer" - have led me to edit this answer. I held off as long as possible because it was intended as a short quip and not as anything "deep" so putting in any explanation seemed counter to the purpose. However, the comments are making it clear that I should be clear to avoid misunderstanding.)
My original answer:
The wording of this part of the specification:
If it's 0, I want to set it to 1, else set it to 0.
implies that the most accurate solution is:
v = dirac_delta(0,v)
First, the confession: I did get my delta functions confused. The Kronecker delta would have been slightly more appropriate, but not by much as I wanted something that was domain-independent (the Kronecker delta is mainly used just for integers). But I really shouldn't have used delta functions at all, I should have said:
v = characteristic_function({0},v)
Let me clarify. Recall that a function is a triple, (X,Y,f), where X and Y are sets (called the domain and codomain respectively) and f is a rule that assigns an element of Y to each element of X. We often write the triple (X,Y,f) as f: X → Y. Given a subset of X, say A, there is a characteristic function which is a function χA: X → {0,1} (it can also be thought of as a function to a larger codomain such as ℕ or ℝ). This function is defined by the rule:
χA(x) = 1 if x ∈ A and χA(x) = 0 if x ∉ A.
If you like truth tables, it's the truth table for the question "Is the element x of X an element of the subset A?".
So from this definition, it's clear that the characteristic function is what is needed here, with X some big set containing 0 and A = {0}. That's what I should have written.
And so to delta functions. For this, we need to know about integration. Either you already know it, or you don't. If you don't, nothing I can say here will tell you about the intricacies of the theory, but I can give a one sentence summary. A measure on a set X is in essence "that which is needed to make averages work". That is to say that if we have a set X and a measure μ on that set then there is a class of functions X → ℝ, called measurable functions for which the expression ∫X f dμ makes sense and is, in some vague sense, the "average" of f over X.
Given a measure on a set, one can define a "measure" for subsets of that set. This is done by assigning to a subset the integral of its characteristic function (assuming that this is a measurable function). This can be infinite, or undefined (the two are subtly different).
There are lots of measures around, but there are two that are important here. One is the standard measure on the real line, ℝ. For this measure, then ∫ℝ f dμ is pretty much what you get taught in school (is calculus still taught in schools?): sum up little rectangles and take smaller and smaller widths. In this measure, the measure of an interval is its width. The measure of a point is 0.
Another important measure, which works on any set, is called the point measure. It is defined so that the integral of a function is the sum of its values:
∫X f dμ = ∑x ∈X f(x)
This measure assigns to each singleton set the measure 1. This means that a subset has finite measure if and only if it is itself finite. And very few functions have finite integral. If a function has a finite integral, it must be non-zero only on a countable number of points. So the vast majority of functions that you probably know do not have finite integral under this measure.
And now to delta functions. Let's take a very broad definition. We have a measurable space (X,μ) (so that's a set with a measure on it) and an element a ∈ X. We "define" the delta function (depending on a) to be the "function" δa: X → ℝ with the property that δa(x) = 0 if x ≠ a and ∫X δa dμ = 1.
The most important fact about this to get a-hold of is this: The delta function need not be a function. It is not properly defined: I have not said what δa(a) is.
What you do at this point depends on who you are. The world here divides in to two categories. If you are a mathematician, you say the following:
Okay, so the delta function might not be defined. Let's look at its hypothetical properties and see if we can find a proper home for it where it is defined. We can do that, and we end up with distributions. These are not (necessarily) functions, but are things that behave a little like functions, and often we can work with them as if they were functions; but there are certain things that they don't have (such as "values") so we need to be careful.
If you are not a mathematician, you say the following:
Okay, so the delta function might not be properly defined. Who says so? A bunch of mathematicians? Ignore them! What do they know?
Having now offended my audience, I shall continue.
The dirac delta is usually taken to be the delta function of a point (often 0) in the real line with its standard measure. So those who are complaining in the comments about me not knowing my deltas are doing so because they are using this definition. To them, I apologise: although I can wriggle out of that by using the Mathematician's defence (as popularised by Humpty Dumpty: simply redefine everything so that it is correct), it is bad form to use a standard term to mean something different.
But there is a delta function which does do what I want it to do and it is that which I need here. If I take a point measure on a set X then there is a genuine function δa : X → ℝ which satisfies the criteria for a delta function. This is because we are looking for a function X → ℝ which is zero except at a and such that the sum of all of its values is 1. Such a function is simple: the only missing piece of information is its value at a, and to get the sum to be 1 we just assign it the value 1. This is none other than the characteristic function on {a}. Then:
∫X δa dμ = ∑x ∈ X δa(x) = δa(a) = 1.
So in this case, for a singleton set, the characteristic function and the delta function agree.
In conclusion, there are three families of "functions" here:
- The characteristic functions of singleton sets,
- The delta functions,
- The Kronecker delta functions.
The second of these is the most general as any of the others is an example of it when using the point measure. But the first and third have the advantage that they are always genuine functions. The third is actually a special case of the first, for a particular family of domains (integers, or some subset thereof).
So, finally, when I originally wrote the answer I wasn't thinking properly (I wouldn't go so far as to say that I was confused, as I hope I've just demonstrated I do know what I'm talking about when I actually think first, I just didn't think very much). The usual meaning of the dirac delta is not what is wanted here, but one of the points of my answer was that the input domain was not defined so the Kronecker delta would also not have been right. Thus the best mathematical answer (which I was aiming for) would have been the characteristic function.
I hope that that is all clear; and I also hope that I never have to write a mathematical piece again using HTML entities instead of TeX macros!
