Is “Dirac Delta” a function?

The short answer is no. The notion of Dirac Delta is that it has value at and value everywhere else, and that its integral is . These characteristics (we will identify these characteristics as (*) later in this article) do not identify with a function.

So, what is Dirac Delta? It is a distribution or a generalized function. Now, what are these objects? We will not go into the precise mathematical definition of a distribution. However, a subset of distributions is generated from integrable functions. And, in fact, there is a one-to-one mapping between locally integrable functions and this subset of distributions. Hence, this subset of distributions can be identified with locally integrable functions – that is, given a locally integrable function, there is one and only one corresponding distribution.

Still, distributions and locally integrable functions are different objects. Nevertheless, they do share a few common characteristics namely:

• Locally integrable functions generate measures (absolutely continuous measures to be specific). Some distributions generate measures (called Radon measures in the context of distributions). Of course, you can use measures to integrate functions.
• Integrable functions can be convolved with each other. Some distributions (dual space of Schwartz space) can be convolved with each other.
• You can the notion of support of an integrable function (in fact, for any function). Likewise, you have the notion of support for distributions.

For these reasons, you can “visualize” certain distributions as functions. You can visualize distribution as being zero everywhere except its support just like a function is zero everywhere except its support. Thus, you can “visualize” the Dirac Delta as being a function with value at and value everywhere else.

However, note that, you can only do permitted operations on distributions (like those listed above). (Also, certain operations are permitted only on a subset of distributions. That is why, we did not get into the mathematical definition of distributions). For example, while you can visualize certain distributions as being functions, it still does not mathematical sense to ask for a value of distribution at a particular point.

Now, the reason you can think of Dirac Delta as having the (*) characteristics is that its precise definition as a distribution owes its origin to the following theorem:

If function is continuous and bounded,

Here, is a set of functions called kernel (or specifically, approximate identify) that have approximate the characteristics (*).

Thus, for a little more mathematical rigor than merely mentioning the characteristics (*), Dirac Delta is mentioned in some undergraduate text books as being limit of a sequence of functions in .

Finally, you may have seen Dirac Delta being used to model a spike in physical situations. That is where the resemblance of Dirac Delta to the intuitive notion of function is useful and that is where it ends. For all subsequent mathematics, it is only used as a distribution – that is, only as part of permitted operations such as convolution.