**Abstract:** Continuity is defined relative to a topology. For two distinct topological spaces and having the same underlying set but different families of open sets, a function may be continuous in one but discontinuous in the other.

The signum function is defined on the real line as follows:

It is the archetype of a function with a jump discontinuity. So how can the title of this article, claiming that such a function may be continuous, be justified? It all depends on the topology!

**Topological Spaces**

A topological space is a pair comprising a set and a family of subsets of . The family must be closed under arbitrary unions and finite intersections, and also contain both and . The sets in are called the open sets. This structure enables us to define continuity and many other concepts, such as dimension, compactness and connectedness.

Metric spaces, in which the distance between each pair of points is defined, are special examples of topological spaces, as the metric may be used to define a topological structure. Euclidean spaces, which are particular metric spaces, are also topological spaces.

A neighbourhood of a point is any open set containing . For the real line , the usual topology is generated by forming arbitrary unions and finite intersections of open intervals. The open ball is a neighbourhood of .

**Continuity**

For two topological spaces and , a function is continuous if the inverse image of every open set is open. That is, for , .

For the usual topology, the – definition of continuity of a function is equivalent to the more general topological definition: the inverse image of every open set is open. For the signum function, it is clear that the inverse image of the open interval is the singleton , which is not an open set. Therefore, the signum function is not continuous with the usual topology. This is no surprise.

Now let us define another topology in . Let be the usual topology on the real line. A strip neighbourhood of is the set . It is infinite in the `vertical’ direction. We generate a topology by forming arbitrary unions and finite intersections of such neighbourhoods. This is the family of all sets of the form where .

It is clear that if is a neighbourhood of then it is also a neighbourhood of for all . Now consider the parametric function from to :

The inverse image of a strip neighbourhood of is an open interval in . This is true even when . Thus, the function, which has the same `graph’ as the signum function defined above, is continuous.

**Separation Axioms**

The bare definition of topology is somewhat barren. The problem of functions that appear discontinuous but that are actually continuous arose because a neighbourhood of a point is also a neighbourhood of any other point in the same `vertical’ line. Two points and cannot be “housed off” from each other.

To remove this difficulty, we need to introduce additional constraints. Felix Hausdorff defined a criterion that solves the problem. We require that any two points in a topological space have disjoint neighbourhoods. Spaces satisfying this condition are now called Hausdorff spaces. There is a hierarchy of conditions known as separation axioms, . Hausdorff’s condition is .

**Sources**

Wikipedia article: Separation axiom.

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