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1 $begingroup$ What are the conditions so that the function defined on the product space $Xtimes Y$ $f: X times Yrightarrow mathbb{Z}$ is continuous. For example, is there a condition that says that if any restriction on $X times{y_0}$ or ${x_0}times Y $ is continuous, then $f$ is continuous? Are there other conditions if we deal with metric spaces ? general-topology continuity share | cite | improve this question edited Jan 3 at 12:32 José Carlos Santos 166k 22 132 235 asked Jul 1 '18 at 16:49