I am a bit confused by certain area of math such as optimization is obsessed with lower semi-continuous (lsc) (and upper semicontinuous) functions, when continuity seems to describe all functions of importance.
Another problem is that the definition of lsc function is extremely hard to remember and online references differ wildly,
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some are phrased in terms of convergent nets,
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others are framed in terms of some epsilon-delta argument involving some strange inequality of the type $f(y) – \epsilon < f(x)$,
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others are framed in terms of liminfs.
They could be equivalent but I have no idea who to trust because I don't have a good intuition of these functions.
Can someone please elucidate the importance of lsc functions and offer a practical definition?
Best Answer
On the definition
In the notes you linked to, Bell defines a function $f$ on a topological space to be lower semi-continuous if, for each $c \in \mathbb{R}$, $\{f > c\}$ is an open set. For my money, this is the most useful definition since it doesn't require first countability or metrizability of the space in question. The definition at the Wikipedia page is actually quite good: I would work through the various equivalent definitions of lower semi-continuity provided there and see if I could make sense of it.
The $\epsilon$-$\delta$ statement of lower semi-continuity relates to the "open set definition" above in exactly the same way as the $\epsilon$-$\delta$ definition of continuity relates to the "open set definition" of continuity. There's really nothing "wild" about it.
An application: Elliptic PDE
I agree, though, that the need or usefulness of semi-continuous functions isn't apparent at first. As someone already pointed out in the comments, semi-continuous functions start showing up during "monotone approximation" as well as optimization. This becomes quite apparent and very useful when studying elliptic partial differential equations.
Suppose we are interested in studying the Poisson equation in a domain $U \subseteq \mathbb{R}^{d}$: \begin{equation*} \left\{ \begin{array}{r l} -\Delta u = f &\text{in} \, \, U \\ u = 0 & \text{on} \, \, \partial U \end{array} \right. \end{equation*} For convenience, assume that $U$ is bounded with smooth boundary and $f$ is bounded and continuous in $U$.
One profitable approach to studying such an equation is the method of sub- and supersolutions. Let's start with $C^{2}$ functions: say that a function $u : \overline{U} \to \mathbb{R}$ is a $C^{2}$ subsolution of the equation if it is continuous in $\overline{U}$, $u \leq 0$ on $\partial U$, $u$ is $C^{2}$ in $U$, and, for each $x \in U$, \begin{equation*} - \Delta u(x) \leq f(x). \end{equation*} One can similarly define $u$ to be a $C^{2}$ supersolution if $u \geq 0$ on $\partial U$ and $-\Delta u(x) \geq f(x)$ for each $x \in U$.
It turns out that if $u_{\text{sub}}$ is a $C^{2}$ subsolution and $u_{\text{super}}$ is a $C^{2}$ supersoluion, then $u_{\text{sub}} \leq u_{\text{super}}$ in $\overline{U}$. This is the maximum principle (or comparison principle). This could seem mysterious at first. A nice way to see what is happening is to observe that, actually, $-\Delta u \mapsto u$ is monotone in the following way: if $u$ and $v$ are $C^{2}$ as above, $u \leq v$ on $\partial U$, and if their Laplacians are strictly ordered, that is, \begin{equation*} -\Delta u(x) < -\Delta v(x) \quad \text{for each} \, \, x \in U \end{equation*} then $u \leq v$ holds in all of $U$. Since the inequality is strict, this is a relatively simple verification: since $u \leq v$ on $\partial U$, the only way the conclusion could fail is if there is a $x_{0} \in U$ such that $u(x_{0}) > v(x_{0})$, in which case we see that $u - v$ attains its maximum at an interior point. Let $x_{M}$ be a point in $U$ where $u - v$ attains its maximum. Since both functions are $C^{2}$, we know that $D^{2}u(x_{M}) - D^{2}v(x_{M}) \leq 0$ and, thus, \begin{equation*} -\Delta u(x_{M}) + \Delta v(x_{M}) \geq 0, \end{equation*} and this contradicts the assumption that $-\Delta u < - \Delta v$ in $U$. (Actually, $-\Delta u \leq -\Delta v$ is sufficient, but the argument isn't as obvious.)
The previous considerations suggest a way of solving our PDE: we could look at all $C^{2}$ functions $u$ satisfying $u = 0$ and $-\Delta u \leq f$ in $U$ and take the one for which $-\Delta u$ is largest. In view of the previous computations, this should simply be the maximum of such functions $u$. In other words, we expect that the solution is $\bar{u}$ given by \begin{equation*} \bar{u}(x) = \sup \left\{ u(x) \, \mid \, u \, \, C^{2} \, \, \text{subsolution of} \, \, - \Delta u \leq f \right\}. \end{equation*} This is a supremum of continuous functions. Therefore, we don't know immediately that it is continuous, only that it is upper semi-continuous. This motivates introducing a notion of subsolution of $-\Delta u \leq f$ for $u$ that are only upper semi-continuous, not $C^{2}$ or even continuous, and a notion of supersolution for lower semi-continuous functions. That is the purview of the theory of viscosity solutions. (It turns out that $\bar{u}$ above is the solution, $\bar{u} = 0$ on $\partial U$, $\bar{u}$ is $C^{2}$ in $U$, and $-\Delta \bar{u} = f$ in $U$.)
I can't give a review of viscosity solutions here. However, I would like to give a taste of why it can be useful. The comparison principle for $-\Delta u = f$ says that if $u$ is an upper semi-continuous viscosity subsolution of $-\Delta u \leq f$ in $U$ with $u \leq 0$ on $\partial U$ and $v$ is a lower semi-continuous viscosity supersolution of $-\Delta v \geq f$ in $U$ with $v \geq 0$ on $\partial U$, then $u \leq v$. In other words, the previous discussion applies to viscosity sub- and supersolutions even if they're not $C^{2}$ or even continuous. (It is beside the point to define viscosity sub- or supersolution here; I use that terminology only so that the interested reader knows the terms to search for.)
One other way to solve $-\Delta u =f$ is to approximate $f$. Suppose that we know that $-\Delta u_{n} = f_{n}$ has a solutions (say, $C^{2}$) $u_{n}$ in $U$ with $u_{n} = 0$ on $\partial U$ and $f_{n} \to f$ uniformly in $\overline{U}$ as $n \to \infty$. It is natural to expect that $u_{n} \to u$ as $n \to \infty$, where $u$ is a (or, hopefully, the) solution of $-\Delta u = f$ in $U$. We saw above that there is a lot of monotonicity around to take advantage of. To that end, let's take not the maximum or minimum of the sequence $(u_{n})_{n \in \mathbb{N}}$, but it's asymptotic max and min --- basically, the limsup and liminf. Define $u^{*}$ and $u_{*}$ by \begin{align*} u^{*}(x) = \lim_{\delta \to 0^{+}} \sup \left\{ u_{n}(y) \, \mid \, n \in \mathbb{N}, \, \, y \in U, \, \, n^{-1} + \|x - y\| < \delta \right\}, \\ u_{*}(x) = \lim_{\delta \to 0^{+}} \inf \left\{ u_{n}(y) \, \mid \, n \in \mathbb{N}, \, \, y \in U, \, \, n^{-1} + \|x - y\| < \delta \right\}. \end{align*} It is a good exercise (that has nothing to do with PDE) to prove that if $(u_{n})_{n \in \mathbb{N}}$ are bounded continuous function in $\overline{U}$, then $u^{*}$ and $u_{*}$ defined as above are respectively upper and lower semi-continuous in $U$. (Part two of the exercise: show that, given a continuous function $\bar{u}$ in $U$, we have $u_{n} \to \bar{u}$ locally uniformly in $U$ if and only if $u^{*} = u_{*} = \bar{u}$ in $U$.)
What is less obvious but extremely useful is $u^{*}$ is a viscosity subsolution of $-\Delta u^{*} \leq f$ in $U$ with $u^{*} \leq 0$ on $\partial U$ and $u_{*}$ is a viscosity supersolution with $u_{*} \geq 0$ on $\partial U$. That is, once again, we see that the monotonicity of $-\Delta$ preserves subsolutions when we maximize and it preserves supersolutions when we minimize, at the expense that the optimization process spits out semi-continuous functions. We can now invoke the comparison principle: it tells us that, necessarily, $u^{*} \leq u_{*}$ in $U$. On the other hand, by definition, $u^{*} \geq u_{*}$ also holds. Therefore, $u^{*} = u_{*}$ in $U$ which implies that $u^{*}$ is actually continuous in $U$ and, furthermore, it tells us that $u_{n} \to u^{*}$ locally uniformly in $U$. Note that this local uniform convergence has come somehow "for free" without us having to prove equicontinuity or even that the pointwise limit exists --- we only made some qualitative claims about $u^{*}$ and $u_{*}$. Finally, since $u^{*}$ is a subsolution and $u_{*}$, a supersolution, the function $u^{*}$ is a solution of $-\Delta u^{*} = f$ in $U$ by definition --- a viscosity solution is nothing more than a function that is both a viscosity subsolution and a viscosity supersolution (by definition).
To sum it up, when studying elliptic partial differential equations like the Poisson equation, it is extremely useful (and eventually very natural) to utilize upper and lower semi-continuous functions --- even if remembering their definition is "extremely hard."