# Definition:Continuous Extension

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## Definition

Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces.

Let $A, B \subseteq S_1$ be subsets of $S_1$ such that $A \subseteq B$.

Let $f: A \to S_2$ and $g: B \to S_2$ be continuous mappings.

Then $g$ is a **continuous extension** of $f$ if and only if:

- $\forall s \in A: f \left({s}\right) = g \left({s}\right)$

That is, a **continuous extension** of $f$ is a continuous mapping on a superset which agrees with $f$ on the domain of $f$.

Simply, it is a continuous mapping which is an extension.

### Real Function

Let $A$, $B \subseteq \R$ be subsets of the real numbers such that $A \subseteq B$.

Let $f: A \to \R$ and $g: B \to \R$ be continuous real functions.

Then $g$ is a **continuous extension** of $f$ if and only if:

- $\forall x \in A : \map f x = \map g x$