Notation for "function from a subset of $X$ into $Y$"? The notation $f: X \to Y$ implies that $X$ is the domain of $f$. It is very regularly convenient to say something like: 

Let $U \subset X$ and $f: U \to Y$. 

but in situations where the actual set $U$ isn't that important, just that it's a subset of $X$. Usually the subset will not be arbitrary, but will be implied by the context. The kind of thing I'm thinking of is: 

Let $\alpha : \mathbb{R} \hookrightarrow \mathbb{R}^{2}$ be a plane curve.

Is there a common choice for notation in this case? I imagine there must be (it might even be $\hookrightarrow$ but I'm not certain, I think that might mean injective). 
The notation does not have to encode that the required subset is connected or open or whatever the situtation might require. That should be implied by context. I only care that it's a subset.

These are called partial functions. Wikipedia suggests the notation f : X ↛ Y or f: X ⇸ Y.  I don't know how standard this notation is, so it would be best to explicitly define what you mean.
Yes, I have usually seen $\hookrightarrow$ reserved for injective functions.
