A type-2 computable real function is necessarily continuous; and this remains true for computations relative to any oracle. Conversely, by the Weierstrass Approximation Theorem, every continuous f : [0; 1]→ R is computable relative to some oracle. In their search for a similar topological characterization of relatively computable multi-valued functions f : [0; 1] ⇒ R (also known as multi-functi...

This paper will give an introduction to the fundamentals of computability theory. Based on Robert Soare’s textbook, The Art of Turing Computability: Theory and Applications, we examine concepts including the Halting problem, properties of Turing jumps and degrees, and Post’s Theorem. In the paper, we will assume the reader has a conceptual understanding of computer programs.