If we have sets $A\subseteq B$, then we call the map $\iota:A\to B, x\mapsto x$ the inclusion map from $A$ to $B$. In functional analysis, we can have such a map between normed vector spaces $X,Y$ and if $\|x\|_Y \leq C\|x\|_X$ for all $x\in X$ we say that the space $X$ embeds continuously into $Y$, which is an important concept for many applications, e.g. PDE.
But in many cases, the inclusion map is not a real inclusion map. If we consider $X=H^1([0,1])$ and $Y=C([0,1])$, then the elements of $X$ are equivalence classes of functions where one representative is a continuous function. So in this case, the inclusion map maps the equivalence class to its existing and unique continuous representative and then this mapping is continuous and we call it the continuous embedding of $X$ into $Y$.
Some time ago I talked with someone about compact embeddings. He asked me what I would call an embedding and I said a continuous, injective map. Then he argued that this is a bad definition because then we would have at least that every Hilbert spaces of the same dimension embed into each other compactly, however the map would be pretty useless. He then suggested that the definition should be at least continuous and useful :P
I know that this is not a precise question but hopefully you get my point.