Kleene defined a continuous realizability over Baire space, i.e. $\mathbb N^{\mathbb N}$ with the product topology. In this model $\forall x\exists y\phi(x,y)$ is valid, if there is a continuous function $f$ such that $\forall x\phi(x,f(x))$ is valid. That sounds like what you are looking for. A realizability model assigns a set of realizers to each formula, and $p\models q$ is valid if there is a suitable partial function mapping realizers of $p$ to realizers $q$. In Kleene's example the realizers are members of Baire space and the functions are the partial continuous ones.

To generalize this example, you should consider the category [*equilogical spaces*][1] as an interesting category to develop such a continuous model theory in. In this category realizers can be points of arbitrary T_0-spaces. I am unsure how much research has been done in this area.

Realizability models can be extended to realizability toposes, if and only if they have *(order) partial combinatory algebras* as object of realizers. This can be gathered from Streicher and Lietz's "Impredicativity entails untypedness" in combination with Hofstra's "All realizability is Relative". In order to get all realized functions to be continuous, I would suggest looking into topological partial combinatory algebras. Some research on these has been done by Ingemarie Bethke.

I invite you to have a look at my PhD. thesis, which can be downloaded from https://research-portal.uu.nl/en/publications/realizability-categories. It contains a lot of information on realizability toposes.

  [1]: http://ncatlab.org/nlab/show/equilogical+space