Let $f$ be a continuous function, bounded and strictly increasing in An open interval. Let $\alpha=\inf f(I), \beta= \sup f(I)$. Prove that $f(I)=]\alpha,\beta[.$ And if $f$ isn't bounded?
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HINT: Suppose that $\alpha<y<\beta$. By the definition of infimum there is an $x\in I$ such that $f(x)<y$. Now use the definition of supremum and the intermediate value theorem to show that $y=f(t)$ for some $t\in I$.
If $f$ isn’t bounded, the intermediate value theorem can still be used to show that $f[I]$ has no ‘holes’.