Let $X$ be a compact topological space and let $f: X \rightarrow \mathbb{R}$ be a function. The graph$f$ is the set $G = \{(x,f(x)) : x\in X\}$. when this $G$ is going to be closed, compact and connected?
By graphically, i can see that if $f$ is continuous. Then G is closed. And by same graphs i guess the converse need not true and G is connected.
My friend told me that if $f$ is bounded and continuous. then G might be compact
But i don't any proof and counterexample for this. i will be happy if i get some help
If $f$ is continuous, then $f[X]$ is compact in the reals and hence bounded. Also, $G$ will then be compact, as the continuous image of $X$ under the continuous map $x \rightarrow (x, f(x))$ from $X$ into $X \times \mathbb{R}$. (Continuous as the compositions wiht the two projections are the identity, resp. $f$, hence continuous by assumption).