Continuous map of pseudotopological spaces

Definition
Suppose $$X, Y$$ are defining ingredient::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 $$\varphi$$ on $$X$$ and any point $$x \in X$$ such that $$\varphi \to x$$, we have:

$$f_*\varphi \to f(x)$$

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

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