TITLE: A function is continuous if its graph is closed
QUESTION [8 upvotes]: Any help with the following problem:
Prove that if a function $f: \mathbb{R}\rightarrow \mathbb{R}$ is bounded and its graph is a closed subset of $\mathbb{R}^{2}$, then $f$ is continuous.

REPLY [2 votes]: Alternative to the answers already presented, one might use sequences. 
Without giving away everything, suppose $x_n \rightarrow x$, and consider the the sequence $f_n:=f(x_n)$. If any subsequence $f_{n_k}$ of $f_n$ converges, the only candidate for it to converge to is $f(x)$ by the closedness of the graph. Considering $f_n$ is bounded, what can be said about $\limsup f_n$ and $\liminf f_n$?