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...

Abstract. A remarkable result due to Kou, Liu & Luo states that the condition of continuity for a dcpo can be split into quasi-continuity and meet-continuity. Their argument contained a gap, however, which is probably why the authors of the monograph Continuous Lattices and Domains used a different (and fairly sophisticated) sequence of lemmas in order to establish the result. In this note we s...

Abstract We introduce notions of concavity for functions on balanced polyhedral spaces, and we show that concave such spaces satisfy several strong continuity properties.

One of the many ground-breaking themes in Evan Thompson’s rich and thoughtprovoking book Mind in Life: Biology, Phenomenology, and the Sciences of Mind is his distinctive development and defence of an idea that he calls the deep continuity of life and mind – henceforth just deep continuity. Thompson introduces this idea as follows: ‚life and mind share a set of basic organizational principles, ...

One of the many ground-breaking themes in Evan Thompson’s rich and thoughtprovoking book Mind in Life: Biology, Phenomenology, and the Sciences of Mind is his distinctive development and defence of an idea that he calls the deep continuity of life and mind – henceforth just deep continuity. Thompson introduces this idea as follows: “life and mind share a set of basic organizational principles, ...