Prove that the graph of continuous function f: $\mathbb{R}\rightarrow\mathbb{R}$ is a connected set

Question

Prove that the graph of continuous function f: $\mathbb{R}\rightarrow\mathbb{R}$ is a connected set in $\mathbb{R}\times\mathbb{R}$ (in euclidean metric).

I'm new to this kind of proofs, so I don't even know how to start it. Any help will be much appreciated.

Answer 1

If the graph is not connected then it has at least two components C_1 and C_2 and we can find two disjoint open sets$O_1$ and $O_2$ in $\mathbb{R}\times\mathbb{R}$ where $C_1 \subset O_1 $ and $C_2 \subset O_2 $

Note that the projection of $O_1$ and $O_2$ on the x-axis are disjoint and cover part of the domain of the function $f$.

That contradicts the continuity of the function $f$

Answer 2

In fact is a path connected set. Just consider the contiuous function $$\gamma(t)=(t,f(t))\text{ for } t\in\Bbb R$$

Then if $A=(a,f(a))$ and $B=(b,f(b))$ are points of the graph ($a<b$) then $\gamma_{a,b}(t)=\gamma(t)$ for $t\in[a,b]$ is a path that connects $A$ and $B$.