Originally Posted by
Drexel28 This is an if and only if. Let $\displaystyle \Gamma_f$ denote the graph, note then that $\displaystyle D=\pi_1(\Gamma_f)$ where $\displaystyle \pi_1:R^2\to R$ is the canonical projection onto the first coordinate--why does this tell us $\displaystyle D$'s connected. Conversely, if $\displaystyle D$ is connected then $\displaystyle \Gamma_f=h(D)$ where $\displaystyle h: D\to R^2: x \mapsto (x,f(x))$--why is $\displaystyle h$ continuous and why does this tell us that $\displaystyle \Gamma_f$ is connected?