Sequential Continuity is Equivalent to Continuity in the Reals

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Theorem

Let $A \subseteq \R$ be a subset of the real numbers.

Let $c \in A$.

Let $f : A \to \R$ be a real function.


Then $f$ is continuous at $c$ if and only if:

for each sequence $\left\langle{x_n}\right\rangle$ in $A$ that converges to $c$, the sequence $\left\langle{f \left({x_n}\right)}\right\rangle$ converges to $f \left({c}\right)$


Proof


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