  Continuity is one of the most central notions in mathematics and related
areas. An interesting related topic is decompositions of continuity, where
continuity is shown to be equivalent to the combination of two or more weak
continuity notions. Now, from a logical viewpoint, continuity is fairly tame in
that its basic properties are readily established in rather weak logical
systems. It is then a natural question whether the same holds for the
aforementioned weak continuity notions. In this paper, we establish the
Janus-face nature of weak continuity notions: we identify numerous weak
continuity notions that are as tame as continuity, but also list a number of
extremely wild weak continuity notions for which basic properties, like finding
the supremum or evaluating the Riemann integral, are extremely hard to prove.
To make our heuristic notions (tame, wild, weak) precise, we adopt Reverse
Mathematics, a program in the foundation of mathematics that seeks to identify
the minimal axioms that prove a given theorem. Particularly surprising and
novel is that our `wild' theorems exist at the outer edges of Reverse
Mathematics, i.e. while rather basic, they imply full second-order arithmetic.
