Computability Theory Vs. Real Analysis
Here's the dirt on how these objects relate to one another:
One of the biggest, least expected connections between these fields comes from the DPRM-Theorem, also known as The Solution to Hilbert's 10th Problem. This theorem says that the recursively enumerable sets are precisely the diophantine sets. This can be used to prove some otherwise intractable statements in real analysis which don't appear to be related to computability theory at all.
In turn, real analysis contributes back to computability theory by providing concepts like "computable real number", "computable real function", and so on, which is collectively the subject of a subfield sometimes called "computable real analysis".
In turn, real analysis contributes back to computability theory by providing concepts like "computable real number", "computable real function", and so on, which is collectively the subject of a subfield sometimes called "computable real analysis".