The definition of a closed function and its epigraph

Question

There is a related discussion:
closed epigraphs equivalence
Showing that projections $\mathbb{R}^2 \to \mathbb{R}$ are not closed

My problem is rather simple:

A function is closed if its epigraph is closed.

For example:

1. $f(x) = x^2$ (also a convex function with its epigraph convex.)
2. $f(x) = \tan^{−1} (x)$

However, I am confused about like the first case, the epigraph approaches infinity as $y\rightarrow \infty$. The second case, the epigraph approaches infinity as $x\rightarrow\infty$. The epigraph has no boundary on the right and left directions.

I cannot understand why their epigraph is closed.

Answer 1

Closed set definition says that: a set is closed when all the limit points of the set are in set. Now if you check even $R$ is a closed set. From this you can get an idea why the above sets(epigraphs of the function) are closed.