Sequential continuity

Definition

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be a function. The function $f$ is sequentially continuous if the following holds. For each sequence $(x_n)$ in $D$ that converges to a point $a \in D$, the sequence $(f(x_n))$ converges to $f(a)$.

Remarks

For real functions, sequential continuity is the same as continuity. The definition is useful to prove that a function is not continuous. It is enough to find one sequence $x_n \to a$ with $f(x_n) \not\to f(a)$. Mathlib defines it for topological spaces, as SeqContinuous.

Explanations

L1 human

A sequentially continuous function preserves limits of sequences. The function of the limit is the limit of the function values.