Let $\pi: X \to X/{\sim}$ be a quotient map and let $Z$ be a topological space. A function $g: X/{\sim} \to Z$ is continuous if and only if $g \circ \pi: X \to Z$ is continuous.
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Consequently, if $f: X \to Z$ is continuous and $f(x) = f(y)$ whenever $x \sim y$, then the unique function $\bar{f}: X/{\sim} \to Z$ satisfying $f = \bar{f} \circ \pi$ is continuous.