I'm a little confused about when the sum of two lower semicontinuous functions is continuous. I couldn't find a neat answer to my question on this site, though there are similar ones.
Say that $f: \mathbb R^n \to [-\infty,\infty]$ is lsc iff $x_n \to x$ implies $$\liminf_n(f(x_n)) \geq f(x).$$
Now suppose $f:\mathbb R^n \to [0,-\infty]$ and $g: \mathbb R^n \to (-\infty,0]$ are both lsc.
Is $f+g$ lsc as well?
An affirmative answer would follow immediately from $$\liminf_n(y_n + z_n) \geq \liminf_n(y_n) + \liminf_n(z_n)\tag{1}$$ for any sequence $(y_n)$ in $[0,\infty]$ and sequence $(z_n)$ in $(-\infty,0]$. For any $m \in \mathbb N$, we have $$y_m + z_m \geq \inf_{n \geq m} y_n + \inf_{n \geq m} z_n,$$ which implies $$\inf_{m \geq n}(y_m + z_m) \geq \inf_{n \geq m} y_n + \inf_{n \geq m} z_n.$$ Then (1) follows by letting $m \to \infty$.
I don't see what's wrong with that argument, and yet the following counterexample seems to hold as well: $$y_n = n, \ \ z_n = -n.$$
Can someone please explain what's going on here?
It seems you're saying that $0 \ge \infty+(-\infty)$ is false. Some might call it true. For any $m$, you have $\inf_{n \ge m} (y_n+z_n) = 0$, $\inf_{n \ge m} y_n = m$, and $\inf_{n \ge m} z_n = -\infty$. So indeed $$\inf_{n \geq m}(y_n + z_n) \geq \inf_{n \geq m} y_n + \inf_{n \geq m} z_n$$ $$0 \ge m+(-\infty)$$ is true. If you take the limit as $m \to \infty$, you'll get $0 \ge \infty+(-\infty)$, which we can say is true for our purposes.
So your question comes down to how you define addition on the extended reals.