I'm trying to proof the following: Let $X$ be a smooth manifold, $X_0$ an open subset of $X$, $i: X_0 \to X$ the canonical inclusion, $Y$ another smooth manifold and $f: Y\to X_0$ continuous, then
$f$ is smooth $\iff i \circ f : Y \to X $ is smooth
I already know that $X_0$ is another smooth manifold and $i$ is smooth. I'm assuming $\Rightarrow$ is an immediate result of the Chain Rule? Though I have no idea where to start the proof for $\Leftarrow$. Any help would be greatly appreciated.
The easiest way is perhaps to use the definition of "smooth" in terms of charts and the knowledge that charts of $X_0$ around $x_0\in X_0$ are of the form $\phi\circ i$ for some chart $\phi$ of $M$ around $i(x_0)$.