What is the most general mathematical object that defines a solution to an ordinary differential equation? What is the most general object that defines a solution to an ordinary differential equation?  (I don't know enough to know if this question is specific enough.  I am hoping the answer will be something like "a function", "a continuous function", "a piecewise continuous function" ... or something like this.)

(All links go to Wikipedia unless stated otherwise.)
Elaborating on joriki's answer: The most general spaces where it does make sense to talk about differential equations are certain classes of  topological vector spaces, it is for example rather straight forward to formulate the concept of a differential equation in Banach spaces. (Here the esolution to an equation is a mapping of topological vector spaces)
For differential equations in $\mathbb{R}^n$, the solutions themselves are elements of certain topological vector spaces.The most general topological vector spaces that are considered are AFAIK Sobolev spaces, these are function spaces such that each individual function is normable with respect to a prescribed $L^p$ norm, has generalized derivatives in the sense of distributions to the order that is necessary to formulate the weak formulation (see also the Azimuth wiki) of the equation one would like to solve, such that the generalized derivatives are again normable with respect to the $L^p$ norm. Some Sobolev spaces have similar characterizations like "all piecwise continuous function that..." which are a little bit more complicated than that, and differ from space to space, so I'd rather like to refer to the extensive literature instead of repeating that here :-(
