TITLE: If $f$ is continuous then $G$ is homeomorphic to $X$.
QUESTION [8 upvotes]: Let $f: X \to Y$ be a function. The graph of $f$ is defined to be the set $G = \{(x, f(x)) : x \in X\}$. Prove that if $f$ is continuous then $G$ is homeomorphic to $X$. 

an anyone suggest  me the process of solving this problem, thanks.

REPLY [4 votes]: Hints: Let $\varphi : x \mapsto (x,f(x))$.

Show that $\varphi$ is a continuous bijection from $X$ to $G$.
Show that the projection on the first coordinate $\text{pr}_1 : X \times Y \to X$ is continuous and $\text{pr}_1 \circ \varphi =\operatorname{Id}_X$.