The Signum Function may be Continuous

Abstract: Continuity is defined relative to a topology. For two distinct topological spaces {(X,\mathcal{O}_1)} and {(X,\mathcal{O}_2)} having the same underlying set {X} 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:

\displaystyle \mathrm{sgn} x = \begin{cases} -1, \quad x < 0 \\ \ \ 0, \quad x = 0 \\ +1, \quad x > 0 \end{cases} \,.

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 {(X,\mathcal{O})} comprising a set {X} and a family {\mathcal{O}} of subsets of {X}. The family {\mathcal{O}} must be closed under arbitrary unions and finite intersections, and also contain both {X} and {\emptyset}. The sets in {\mathcal{O}} 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 {x\in X} is any open set containing {x}. For the real line {\mathbb{R}}, the usual topology is generated by forming arbitrary unions and finite intersections of open intervals. The open ball {B_\varepsilon(x) = \{x : |x-x_0|<\varepsilon \}} is a neighbourhood of {x_0}.

Continuity

For two topological spaces {(X,\mathcal{O}_X)} and {(Y,\mathcal{O}_Y)}, a function {f:X\rightarrow Y} is continuous if the inverse image of every open set is open. That is, for {U\in \mathcal{O}_Y}, {f^{-1}(U)\in\mathcal{O}_X}.

For the usual topology, the {\varepsilon}{\delta} definition of continuity of a function {f:\mathbb{R}\rightarrow\mathbb{R}} 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 {(-1,+1)} is the singleton {\{0\}}, 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 {\mathbb{R}^2}. Let {\mathcal{O}_1} be the usual topology on the real line. A strip neighbourhood of {(x_0,y_0)} is the set {\{(x,y): |x-x_0|<\varepsilon\ \mbox{and}\ y\in\mathbb{R}\}}. 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 {\mathcal{O}_2} of all sets of the form {U\times\mathbb{R}} where {U\in\mathcal{O}_1}.

It is clear that if {U} is a neighbourhood of {(x_0,y_0)} then it is also a neighbourhood of {(x_0,y)} for all {y\in\mathbb{R}}. Now consider the parametric function from {\mathbb{R}} to {\mathbb{R}^2}:

x(t) = t \,, \qquad y(t) = \mathrm{sgn}\ t \qquad\mbox{for}\qquad t\in\mathbb{R} \,.

The inverse image of a strip neighbourhood of {(x,y)} is an open interval in {\mathbb{R}}. This is true even when {x=0}. 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 {(x,y)} is also a neighbourhood of any other point in the same `vertical’ line. Two points {(x,y_1)} and {(x,y_2)} 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 {(X,\mathcal{O})} have disjoint neighbourhoods. Spaces satisfying this condition are now called Hausdorff spaces. There is a hierarchy of conditions known as separation axioms, {\mathrm{T}_0, \mathrm{T}_1, \mathrm{T}_2, \mathrm{T}_3, \dots\ }. Hausdorff’s condition is {\mathrm{T}_2}.

Sources

{\bullet} Wikipedia article: Separation axiom.

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