I'm taking some maps from quotient spaces to prove the continuity. I'm thinking if I can use the characteristic property to prove the continuity of the map. For example, take the map $f:\mathbb R/\sim \to \mathbb S^1$, where the equivalence relation is given by $x\sim y$ iff $x$ and $y$ differ by an integer, so I take this map: $f:\mathbb R/\sim \to \mathbb S^1$ defined by $f([x])=(\cos(2\pi x),\sin(2\pi x))$. So the question is if I prove that this map $g:\mathbb R \to \mathbb S^1$, $g(x)=(\cos(2\pi x),\sin(2\pi x))$ is continuous, then $f$ is continuous? Please anyone can help?