The negative entropy function is: \begin{align*} f(x) = \begin{cases} \sum_{i=1}^n x_i \log(x_i), & x \in \mathbb{R}^{n}_{++}\\ \infty, & \text{ otherwise}. \end{cases} \end{align*}
I know how to show that this is a convex function (this is the easy part). Moreover, by Wikipedia, we see that a function is a proper convex function if $f(x) >-\infty$ for all $x$ and $f(x) <\infty$ for at least one $x$. The first condition $f(x) >-\infty$ can be seen by minimizing the terms $x_i \log(x_i)$ which gives $-1/e$ hence the summation is at least $-n/e > -\infty$. The latter condition $f(x) < \infty$ is easy to see. Now I just want to conclude that this is also closed.
By Wikipedia, "A proper convex function is closed if and only if it is lower semi-continuous." So, should I just show $f(x)$ is lower semi-continuous? If yes, how?