Connected Subsets of X x Y Let $X$ be a connected topological space and $f : X\to Y$ a map. Show that the graph $G(f)$,
defined by $G(f) = \{(x; f(x)) \in X \times  Y | x \in X\}$ is a connected subset of $X \times  Y$. 
I know that the image of a connected space under a continuous map is connected, so Y is connected; however, I'm not sure how to prove that G(f) is connected. It seems somewhat obvious to me. 
 A: As you already know,$f(X)$ is connected. In order to prove that the product $X\times f(X)$ is connected use the fact that a topological space $A$ is disconnected if there is an onto continuous function $g:A\to ${$0,1$} with the discrete topology.
A: Hint: Use the universal property of products. Since each coordinate map is continuous, your map to the product is continuous. Use the fact that $\text{id}:X\to X$ is continuous, and $f:X\to Y$ is continuous. 
A: Let me suggest a slightly less "beat on it with fancy machinery" approach that that suggested by LASV. Let
$$
h : X \rightarrow G(f) : x \mapsto (x, f(x)).
$$
Then $h$ is evidently continuous. Now suppose that $G(f)$ were disconnected, so $G(f) = U \cup V$ where $U$ and $V$ are nonempty disjoint open sets. Can you use $h$ to produce nonempty disjoint open sets in $X$ whose union is all of $X$? 
A: If you know that the image of a connected space under a continuous map is connected, apply this to the map $x\mapsto (x,f(x))$. As to why this map is continuous: If John's “evidently” does not suffice, look at LASV's answer.
