Solution verification: a continuous function in an open interval $(a,b)$ with finite limits of $\lim_{x \rightarrow a^+}f(x)$ and $\lim_{x \rightarrow b^-}f(x)$ is bounded
Proof: $\lim_{x \rightarrow a^+}f(x)$ and $\lim_{x \rightarrow b^-}f(x)$ exist and are finite. Lets denote them: $L_1$ and $L_2$ respectively. So since $f$ is continuous there are $c,d \in (a,b)$ so that $f(c)=L_1$ and $f(d)=L_2$. So since $f$ is continuous in $[c,d] \subseteq (a,b)$, then it's bounded there, so it must also be bounded in $(a,b)$.
Is this solution correct?
No, it's not: on an open interval, limits are not necessarily attained. But you can extend $f$ to a continuous function $\tilde f$ on the closed, bounded interval $[a,b]$. Now it's a theorem that a continuous function on a closed bounded interval is bounded on that interval (and actually attains a maximum and a minimum) (Weierstrass's boundedness theorem). So $f$, which is the restriction of $\tilde f$ to the open interval $]a,b[$, is also bounded (but doesn't necessarily attains a maximum nor a minimum on $]a,b[$.