(Theorem) Continuity of maps relative to the product topology

Munkres 19.6: Continuity of maps relative to the product topology

Given an indexed family of sets $\{X_{\alpha }{\}}_{\alpha \in J}$, let the map to their generalized product $f:Z\to {\prod }_{\alpha \in J}{X}_{\alpha }$ be given by

$$f(z)=(f_{\alpha }(z){)}_{\alpha \in J},$$

where $f_{\alpha }:Z\to X_{\alpha }$ is the coordinate function $f_{\alpha }:{\pi }_{\alpha }\circ f$. If ${\prod }_{\alpha \in J}{X}_{\alpha }$ has the product topology, then $f$ is continuous if and only if each $f_{\alpha }$ is continuous.