If $f$ is continious ,then $G$ is connected ? True/false

Question

let $X$ be a compact topological space and let $f : X \rightarrow \mathbb{R}$ be function .The graph of $f$ is the set $G = \{(x,f(x)) : x \in X \} \subseteq X \times \mathbb{R}$.

Now my Question is that

If $f$ is continious ,then $G$ is connected ? True/false

i thinks it will be true because continious image of connected set is connected

Any hints/solution

thanks u

Answer 1

Consider $X=[0,1]\cup[2,3]$ and define $f:X \to \Bbb{R}$ by

$f(x) = x$ if $x \in [0,1]$ and $f(x)=x-1$ if $x \in [2,3]$.

Then by Pasting lemma $f$ is continuous but has disconnected graph.

Graph of $f$: