Prove or provide an example where this is false: There exists $f: (0,1] \rightarrow R $ that is continuous and "onto" The idea is to explain how this is true, or prove an example that disproves.
There exists $f: (0,1] \rightarrow R$ that is continuous and "onto."
My question is: how do I get in the mindset of finding examples that prove or disprove questions like this? 
(I know that in order to prove that this is true, I need to use the Intermediate value theorem.)

Try analysing this function's graph $f(x)=\frac 1 x \sin \frac 1 x, x \in (0, 1]$.
