(a) Let $f: X \rightarrow Y$ be a continuous surjection of topological spaces. Prove that, if $X$ is connected, then $Y$ is also connected.

(b) Let $g:[0,1] \rightarrow[0,1]$ be a continuous map. Deduce from part (a) that, for every $y$ between $g(0)$ and $g(1)$, there is $x \in[0,1]$ such that $g(x)=y$. [You may not assume the Intermediate Value Theorem, but you may use the fact that suprema exist in $\mathbb{R}$.]