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.

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Sequence definition
Real sequence definition
Limit of a sequence definition
Sequential continuity definition

Tag	`0MVDQEAB60`
URI	`m2://0MVDQEAB60`
Kind	definition
Writing	W0 stub
MSC	26A15
Defines	Sequential continuity
Uses	Limit of a sequence

Sequential continuity

Definition

Remarks

Reading path

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