Any hint on how to approach this problem?
Thanks as always
Let $\displaystyle g:[a,b]\longrightarrow\Gamma_f$ be $\displaystyle g(x)=(x,f(x))$. Let $\displaystyle \{C_i\}$ be an open cover of $\displaystyle \Gamma_f$. The continuity of $\displaystyle g$ implies that $\displaystyle g^{-1}(C_i)$ is open in $\displaystyle [a,b]$ (this is a theorem) and $\displaystyle \{g^{-1}(C_i)\}$ is an open cover of $\displaystyle [a,b]$.
Spoiler:
You want to approach connectedness similarly. Assume that $\displaystyle \Gamma_f\subset U\cup V$ where $\displaystyle U$ and $\displaystyle V$ are open sets such that $\displaystyle cl(U)\cap V=U\cap cl(V)=\emptyset$ and try to derive a contradiction. (Your contradiction will be that $\displaystyle [a,b]$ is not connected.)