Let $f:A\rightarrow\prod_{\alpha\in J} X_\alpha$ be given by the equation $f(a)=(f_\alpha (a))_{\alpha \in J}$ where $f_{\alpha}:A\rightarrow X_\alpha$ for each $\alpha$. Let $\Pi X_\alpha$ have the box topology. Show that the implication; "the function $f$ is continuous if each $f_\alpha$ is continuous" is not true for this topology. How do I prove this? Can anyone help?
Obviously this is true for the product topology (Munkres, Thm 19.6), but I can't figure out why it is not true for the box.