Pointwise Continuity

Context

Statements

Theorem (@Men90, Theorem 5.4). Let $(X, d_{X})$ and $(Y, d_{Y})$ be metric spaces. A function $f : X \to Y$ is continuous at a point $x \in X$ if and only if, for a convergent sequence ${x_{i}}_{i \in N} \subset X$ , the sequence ${f (x_{i})}_{i \in N}$ is also convergent.