# Real Function is Continuous at Isolated Point

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## Theorem

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

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

Let $x \in A$ be an isolated point of $A$.

Then $f$ is continuous at $x$.

## Proof

limit in this case is trivially equal to $f \left({x}\right)$.