Topology -- Continuity and the induced topology. Here is my question. 

Let $X=Y=\Bbb R$, with the usual topology. Let $A=[0,1]$ and topologize $A$ with the induced topology from $X$. 

Does there exist a continuous function from the topological space $A$ onto $Y$? 

Why or why not?

I'd argue No, since we'd be going from a closed to open set. But I also thought that any subset of a topology is an open subset. 

Any help would be appreciated.
No: $[0,1]$ is compact, $\mathbb R$ is not compact, and the image of a compact set under a continuous function is always compact.
The extreme value theorem, from real analysis, states that any continuous function from a compact set to the reals is bounded (and has a maximum). A bounded function cannot be surjective onto $\Bbb R$.