*When I started to write this question I was really confused, but now I think I am starting to get it. Nonetheless I'd like some confirmation that my understanding is correct.*

~~I have read, heard said, and even said myself that in Bishop-style constructive mathematics all functions are continuous---even computable.~~ (**Update: This last sentece was poorly worded. See Carl's correction below.**) (This isn't an axiom or a constructively provable fact. It just is that one cannot constructively prove the existence of a discontinuous function.) I am starting to realize this is not quite as straight-forward as I thought, so I'd like some clarification.

The issue seems to arise when one defines a function $f:A \rightarrow \mathbb{R}$ where $A$ is a subset of a set $X$. (Bishop understands a *subset* to be an (computable) imbedding $i: A \hookrightarrow X$.) To complicate matters Bishop calls the space of such functions $\mathcal{F}(X)$ making them sound as if they should be understood as partial (and therefore partial computable!) functions on $X$. However, it seems such functions $f$ are not always continuous on $X$ or even on $A$ when $A$ is given the restricted topology of $X$. Instead it seems that $f$ is only sure to be continuous on the topology given by the representation of $A$, which can differ greatly from the topology on $X$.

**My questions:**

**Does my understanding seem correct?****Are there any good resources on this subject?**

Here are a few examples of the phenomena I am talking about.

Given a "complemented set" $A=(A_0,A_1)$ where $A_0,A_1 \subseteq X$ are disjoint, e.g. the pair $[0,1/2], (1/2,1]$, then Bishop and Bridges (p 73) construct the characteristic function as $$\chi_A(x) = \begin{cases} 1 & x \in A_0, \\ 0 & x \in A_1. \\ \end{cases}$$ with domain $A_0 \cup A_1$. This function is continuous (even computable) on the disjoint sum of $A_0$ and $A_1$, but not necessarily on $A_0 \cup A_1$ (using the topology of $X$). Since it is not constructively provable that, say, $[0,1/2] \cup (1/2,1] = [0,1]$, it is therefore not constructively provable that $\chi_{[0,1/2]}$ is a discontinuous function on $[0,1]$.

Given a sequence of functions $\phi_n : X \rightarrow \mathbb{R}$, Bishop and Bridges (p 225), define the function $\phi = \sum_n \phi_n$ with domain $$\{x \in X : \sum_n |\phi_n(x)|\ \text{ converges}\}.$$ (The context is measure theory.) It seems that for this definition to make sense constructively, the domain would have to be understood via its $\Pi^0_3$ definition and is therefore represented as a countable intersection of countable unions of closed sets. Then for any $x$ in the set, from each representation of $x$ one can uniformly compute $\sum_n |\phi_n(x)|$ and therefore compute $\sum_n \phi_n(x)$.