In this presentation by Martín Escardó I came across a nice concept of continuity in the set of all functions .

For and we say that

if and only if for all we have .

Then we define to be **finitely continuous** if

So if you want to know the first positions of the output sequence , it suffices to know the first positions of the input sequence .

It turns out that this definition is equivalent to topological equivalence of functions , where is endowed with the discrete topology and carries the interval topology.

Nice, isn’t it?