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# 1.9: Continuity

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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.