📂MetricSpace

Equivalent Conditions for a Function to Be Continuous in a Metric Space

Theorem 1

For two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$, suppose that $E\subset X$ and $p \in E$, and $f : E \to Y$. Then, the following three propositions are equivalent.

(1a) $f$ is continuous at $p$.

(1b) $\lim \limits_{x \to p} f(x)=f(p)$.

(1c) For $\lim \limits_{n\to\infty} p_{n}=p$ that is $\left\{ p_{n} \right\}$, $\lim \limits_{n\to\infty} f(p_{n})=f(p)$.

Proof

• (1a) $\iff$ (1b)

  This is trivial by the definition of limit and continuity.

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• (1b) $\iff$ (1c)

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