There are several ways in which continuity can be formulated as two conditions - in a way such that one of them is lower semicontinuity and the other one is upper semicontinuity. (See below for examples.)
What is a good way to remember which is which? Is there a natural way looking at this which tells me that this one should be called lower and the other one upper semicontinuity?
If there is some good way to remember several of the conditions at the same time, even better. Or is a better strategy simply to remember some typical example of lower/uppser semicontinuos function - and when in doubt, simply to check which of the two parts holds for this function?
Let us work with a function $f\colon\mathbb R\to\mathbb R$. (Although all of this can be easily generalized to the case when domain is arbitrary topological space.)
Here are conditions which are equivalent to continuity of $f$ and I also rewrote them into two conditions (with lower semicontinuity on the left)
- The function $f$ is continuous at $x$ if for every $\varepsilon>0$ there exists $\delta>0$ such that for $|y-x|<\delta$ we have $|f(y)-f(x)|<\varepsilon$, i.e., $$f(x)-\varepsilon< f(y) \qquad\text{and}\qquad f(y)<f(x)+\varepsilon.$$
- The function $f$ is continuous at $x$ if $\lim\limits_{t\to x} f(t)=f(x)$, i.e., $$f(x) \le \liminf_{t\to x} f(t) \qquad\text{and}\qquad \limsup_{t\to x} f(t)\le f(x).$$
- The function $f$ is continuous on $\mathbb R$ if for any $a<b$ the set $f^{-1}(a,b)=\{x\in\mathbb R; a<f(x)<b\}$ is open, i.e., both $$f^{-1}(a,\infty)=\{x\in\mathbb R; a<f(x)\} \qquad\text{and}\qquad f^{-1}(\infty,b)=\{x\in\mathbb R; f(x)<b\}$$ are open.