1.9: Continuity

The angle measure is also assumed to be continuous. Namely, the following property of angle measure will become a part of the axioms:

The function

\[\measuredangle: (A, O, B) \mapsto \measuredangle AOB\]

is continuous at any triple of poitns \((A, O, B)\) such that \(O \ne A\) and \(O \ne B\) and \(\measuredangle AOB \ne \pi\).

To explain this property, we need to extend the notion of continuity to the functions between metric spaces. The definition is a straightforward generalization of the standard definition for the real-to-real functions.

Further, let \(\mathcal{X}\) and \(\mathcal{Y}\) be two metric spaces, and \(d_{\mathcal{X}}, d_{\mathcal{Y}}\) be their metrics.

A map \(f: \mathcal{X} \to \mathcal{Y}\) is called continuous at point \(A \in \mathcal{X}\) if for any \(\varepsilon > 0\) there is \(\delta > 0\), such that

\(d_{\mathcal{X}} (A, A') < \delta \Rightarrow d_{\mathcal{Y}}(f(A), f(A')) < \varepsilon.\)

(Informally it means that sufficiently small changes of \(A\) result in arbitrarily small changes of \(f(A)\).)

A map \(f: \mathcal{X} \to \mathcal{Y}\) is called continuous if it is continuous at every point \(A \in \mathcal{X}\).

One may define a continuous map of several variables the same way. Assume \(f(A, B, C)\) is a function which returns a point in the space \(\mathcal{Y}\) for a triple of points \((A, B, C)\) in the space \(\mathcal{X}\). The map \(f\) might be defined only for some triples in \(\mathcal{X}\).

Assume \(f(A, B, C)\) is defined. Then, we say that \(f\) is continuous at the triple \((A, B, C)\) if for any \(\varepsilon > 0\) there is \(\delta > 0\) such that

\(d_{\mathcal{Y}} (f(A, B, C), f(A',B',C')) < \varepsilon.\)

if \(d_{\mathcal{X}} (A, A') < \delta, d_{\mathcal{X}} (B, B') < \delta\), and \(d_{\mathcal{X}} (C, C') < \delta\).

Exercise \(\PageIndex{1}\)

Let \(\mathcal{X}\) be a metric space.

(a) Let \(A \in \mathcal{X}\) be a fixed point. Show that the function
\[f(B):= d_{\mathcal{X}} (A, B)\]
is continuous at any point \(B\).

(b) Show that \(d_{\mathcal{X}} (A, B)\) is continuous at any pair \(A, B \in \mathcal{X}\).

Hint

(a). By the triangle inequality, \(|f(A') - f(A)| \le d(A', A)\). Therefore, we can take \(\delta = \varepsilon\).

(b). By the triangle inequality,

\[\begin{array} {rcl} {|f(A',B') - f(A, B)|} & \le & {|f(A',B') - F(A, B')| + |F(A, B') - F(A, B)|} \\ {} & \le & {d(A',A) + d(B',B).} \end{array}\]

Therefore, we can take \(\delta = \dfrac{\varepsilon}{2}\).

Exercise \(\PageIndex{2}\)

Let \(\mathcal{X}, \mathcal{Y}\), and \(\mathcal{Z}\) be metric spaces. Assume that the functions \(f: \mathcal{X} \to \mathcal{Y}\) and \(g: \mathcal{Y} \to \mathcal{Z}\) are continuous at any point, and \(h = g \circ f\) is their composition; that is, \(h(A) = g(f(A))\) for any \(A \in \mathcal{X}\). Show that \(h: \mathcal{X} \to \mathcal{Z}\) is continuous at any point.

Hint

Fix \(A \in \mathcal{X}\) and \(B \in \mathcal{Y}\) such that \(f(A) = B\).

Fix \(\varepsilon > 0\). Since \(g\) is continuous at \(B\), there is a positive value \(\delta_1\) such that

\(d_{\mathcal{Z}} (g(B'), g(B)) < \varepsilon\) if \(d_{\mathcal{Y}} (B', B) < \delta_1\).

Since \(f\) is continuous at \(A\), there is \(\delta_2 > 0\) such that

\(d_{\mathcal{Y}} (f(A'), f(A)) < \delta_1\) if \(d_{\mathcal{X}} (A', A) < \delta_2\).

Since \(f(A) = B\), we get that

\(d_{\mathcal{Z}} (h(A'), h(A)) < \varepsilon\) if \(d_{\mathcal{X}} (A',A) < \delta_2\).

Hence the result.

Exercise \(\PageIndex{3}\)

Show that any distance-preserving map is continuous at any point.