= Continuous image of a real interval
A <continuous function> on any <real interval> has image a <real interval>. Given two image values and an intermediate value, choose preimages and apply the <intermediate value theorem> on the closed segment joining them, which stays inside the domain. Neither compactness nor boundedness of the original interval is needed. In particular $f(x)=\sin(1/x)/x$ on $(0,1]$ is a continuous surjection onto $\mathbb R$: its values are unbounded above and below, so its interval image contains every real number.
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