# Sequential Continuity is Equivalent to Continuity in the Reals

Jump to navigation
Jump to search

## 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

## Sources

- 2011: Robert G. Bartle and Donald R. Sherbert:
*Introduction to Real Analysis*(4th ed.): $\S 5.1$: Theorem $2$