Families of functions closed under integration

Another example (which is quite useful for a number of reasons) is the set of trigonometric polynomials

$$f(x) = \sum\limits_{k \in A} c_k e^{i k x}$$

where $A \subseteq \mathbb{Z}$ is finite. This is both closed under integration and differentiation, and in fact has some nice density properties related to Fourier series.

To make the functions strictly real valued, we would consider the (essentially identical) family

$$f(x) = \sum\limits_{k \in A} a_k \sin{kx} + b_k \cos{kx}$$

Two rather trivial examples:

• All polynomial functions divisible by $x^{2013}$ (every one has exactly one primitive in the class, and the class is not closed under differentiation).
• $\{\, x\mapsto ce^x \mid c\in \Bbb R\,\}$ (differentation or taking primitives is rather boring on this set). Since you did not ask for a vector space, you can also let $c$ range over a subset of$~\Bbb R$.