What is the role of continuity in definition of distributions?

Question

Why one can not say simply that a distribution is a linear functional on space $\mathcal D$? What is the purpose or application of topology here?

Answer 1

If we remove continuity, the space $\mathcal{D}'$ of distributions certainly would not be the same:

The existence of linear functionals which are discontinuous on $\mathcal{D}$ may be demonstrated mathematically using the axiom of choice. However, no explicit example of these can be cited and there is very little chance of ever meeting one in practice. (Source)

In general, smaller the topology on $\mathcal{D}$, bigger the dual $\mathcal{D'}$. The topology Schwartz put in $\mathcal{D}$ is precisely the one that yields the dual with no more elements than those needed to ensure that each continuous function is infinitely differentiable:

ainsi nous avons introduit le moins possible d'êtres mathématiques nouveaux pour que toute fonction continue devienne indéfiniment dérivable. (Source)

I think this can be seen as a kind of "role" of continuity (it produces an "ideal amount" of distributions).