Continuous map of pseudotopological spaces

Definition

Suppose $X,Y$ are pseudotopological spaces (with the usual abuse of notation of identifying a pseudotopological space with its underlying set). Suppose $f:X\to Y$ is a function. We say that $f$ is a continuous map of pseudotopological spaces if for any ultrafilter $\phi$ on $X$ and any point $x\in X$ such that $\phi \to x$, we have:

$f_{*}\phi \to f(x)$

In other words, the pushforward of $\phi$ converges to the image of $x$. Here, we define $f_{*}\phi$ as follows:

$f_{*}\phi =\{A\mid f^{-1}(A)\in \phi \}$.