# Can differentiation be thought of as a function?

Would it be wrong to think of differentiation as a function whose domain is the set of differentiable functions and co-domain is the set of all functions whose domain and range are some subsets of real numbers?

• Differentiation is an operator - it transforms a differentiable function into another function. – Mark Viola Apr 26 '17 at 5:14
• – Martín-Blas Pérez Pinilla Apr 26 '17 at 7:21
• Differentiation can be viewed as a function in the sense you describe. Note carefully, however, that the image of the differentiation operator is (tautologically) the set of all derivatives, a complicated space; see for example How discontinuous can a derivative be? – Andrew D. Hwang Apr 26 '17 at 10:51
• @AndrewD.Hwang well, it's not necessary to know the exact image of the operator. As long as you know a co-domain, you can use the operator. Knowing the image is important when you want to consider inverting the operator, but there it would make sense to just take the integrable functions. – leftaroundabout Apr 26 '17 at 12:46
• @leftaroundabout: That's why I left a comment, not an answer. ;) The important (and possibly surprising, and probably of interest to the OP) point is, the image of differentiation is not a simple, familiar function space. (Incidentally, the space of integrable functions is not a suitable domain for an inverse: Many integrable functions do not satisfy the intermediate value property and so are not derivatives. Separately but maybe also relevant, many derivatives are unbounded, and consequently not integrable.) – Andrew D. Hwang Apr 26 '17 at 21:11

Yes, differentiation can be thought of as a function from the set of differentiable functions to the set of functions which are derivatives of a differentiable function. (which is, as Dr. MV points out in a comment, not quite the set of integrable functions).

Such things, which map functions to functions, are typically called operators, but this is just a convention, you can think of them as functions just fine.

Differentiation of a function is a linear operator, a function on a set of functions.

Differentiation at a point is a linear functional, a function which maps elements of a vector space to it's underlying field, in this case functions to real numbers.

• This is a solid answer! (+1) – Mark Viola Apr 26 '17 at 5:37
• Solid answer except that I feel like if you tell some people that something is a "function" when it's really a functional, they would get annoyed. :\ +1 nonetheless... – user541686 Apr 26 '17 at 7:56
• @Mehrdad: A functional is a function.I would hope that anyone annoyed by that would think hard about what exactly is bothering them so much. – Alex Jones Apr 26 '17 at 8:02

Look up differential forms. I think you'll find that what you're describing closely resembles them. In fact, any course on differential topology or differential geometry should present this perspective.

Differentiation can indeed be thought of as a function, but it was not viewed that way by the creators of the calculus. The first mathematician to think of differentiation as an operator from functions to functions was Euler's successor Lagrange.