📂MetricSpace

The Composition of Continuous Functions in Metric Spaces Preserves Continuity

Theorem

Let there be three metric spaces $(X,d_{X})$, $(Y,d_{Y})$, $(Z,d_{Z})$. Assume that $E\subset X$ and there are two functions $f:E\to Y$, $g:f(E) \to Z$. Also, let $h : E \to Z$ defined in $E$ be as follows.

$$h(x) = g(f(x))\quad \forall x \in E$$

If $f$ is continuous at $p\in E$ and $g$ is continuous at $f(p)\in f(E)$, then $h$ is also continuous at $p$. Here, $h$ is called the composition of $f$ and $g$ and is expressed as $h=g\circ f$.

Proof

Let any positive number $\varepsilon>0$ be given. Assuming that $g$ is continuous at $f(p)$, for $\varepsilon$

$$y\in f(E) \quad \text{and} \quad d_{Y}(y,f(p)) < \delta \implies d_{Z}(g(y),g(f(p))) <\varepsilon$$

there exists a positive number $\delta >0$. Then, assuming that $f$ is continuous at $p$, for such $\delta$

$$x \in E \quad \text{and} \quad d_{X}(x,p) <\eta \implies d_{Y}(f(x),f(p))<\delta$$

there exists a positive number $\eta>0$. Therefore, for any positive number $\varepsilon$

$$x\in E \quad \text{and} \quad d_{X}(x,p) <\eta \implies d_{z}(h(x),h(p))=d_{z}(g(f(x)),g(f(p)))< \varepsilon$$

there exists $\eta>0$ such that $h$ is continuous at $p$ by the definition of continuity.

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