**Warning:** The approach described below does not work. The problem is the pointwise addition on the continuous partial functions is not the same operation as pointwise addition on the musco maps; the domain of the results in the former case can be a strict subset of the set of single-valued values of the latter. As a consequence, we don't get the group structure, as pointed out by Tarakh in the comments.

*Side remark: I am reasoning about this natively in the setting of represented spaces, not of topological spaces. There is a chance that to translate properly, some sequentializations need to be added in some places.*

I would start by looking at the space $\mathcal{P}(X,\mathbb{R})$ of continuous partial functions from $X$ to $\mathbb{R}$. These naturally have $G_\delta$-domains (not necessarily dense though), and pointwise addition is a continuous operation.

The topology on this space is found as follows: Fix a dense sequence $(a_n)_{n \in \mathbb{N}}$ in $X$ and a universal Turing function $\Phi$. We say that $x \in 2^\omega$ codes $f : \subseteq \mathbf{X} \to \mathbb{R}$ iff whenever $p \in \omega^\omega$ is such that $d(a_{p(i)},x) < 2^{-i}$ for some $x \in X$, then if $x \in dom(f)$, $\Phi^x(p)$ is total, and $\Phi^x(p)(n)$ codes a rational $q$ with $d(q,f(x)) < 2^{-n}$, and if $x \notin dom(f)$, then $\Phi^x(p)$ is partial. Take the subspace of $2^\omega$ of those $x$ that do code a function, and quotient by coding the same function.

Assuming that we understand the space of upper semicontinuous maps with compact images to be $\mathcal{C}(\mathbf{X},\mathcal{K}(\mathbb{R}))$ (where $\mathcal{K}(\mathbb{R})$ essentially carries the topology induced by $\{\{K \mid K \subseteq U\} \mid U \in \mathcal{O}(\mathbb{R}))$, then we have that:

**Proposition**: $\operatorname{single} : \mathcal{C}(\mathbf{X},\mathcal{K} (\mathbb{R})) \to \mathcal{P}(\mathbf{X},\mathcal{R})$ is continuous.

The proof is essentially noting that we can continuously extract a point from a compact singleton.

We could then just look at the subspace of $\mathcal{P}(X,\mathbb{R})$ arising as the image of the musco's under that function to obtain a candidate for the desired topology. This has the feature that the total continuous functions $\mathcal{C}(\mathbf{X},\mathbb{R})$ show up as a subspace with the usual topology.

This topology is a quite weak one, though, and the natural reversal operation of taking the closure of the graph of a partial function is not continuous with any desirable topology on the right hand side.