The definition of a closed function and its epigraph 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. ![enter image description here](

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.