I need to show that $$f\colon X/{\sim} \to Y \text{ is continuous} \iff \pi\circ f\colon X \to Y \text{ is continuous}$$
where $X/{\sim}$ is a quotient topology and $\pi$ is the quotient map.
I understand the proof for $$f\colon X/{\sim} \to Y \text{ is continuous} \iff f\circ \pi\colon X \to Y \text{ is continuous}$$ which is quite simple. But I can't see how to break this one down.
A general hint: by definition of the quotient topology, the projection $\pi$ is continuous. Question: is the composition of continuous functions a continuous function?
In your case I do not agree with notation, as $\pi$ is a quotient map, i.e. $\pi: X\rightarrow X\setminus\sim$ or $\pi: Y\rightarrow Y\setminus\sim$ . In either the first or second case, I can not compose $\pi$ with $f: X\setminus\sim\rightarrow Y$ and arrive at the map $\pi\circ f: X\rightarrow Y$.