Semicontinuity for single variable function. What is it all about?

[perwiradua]

Let $f: X \rightarrow \mathbb{R}$. The definition given for upper semicontinuity at a point $a$ is for any positive $\epsilon$ there is a positive $\delta$ such that if $|x - a| < \delta$ then $f(x) < f(a) + \epsilon$.

My understanding is, for ordinary continuity at the point $a$ then the inequality $|f(x) - f(a)| < \epsilon$ or $f(a) - \epsilon < f(x) < f(a) + \epsilon$ holds. Therefore for upper semicontinuity we are interested in the right part of the inequality (for lower semicontinuity, we are interested in the left part).

For example the graph
$$f(x) = \left \{ \begin{array}{cc} x^2, & x \neq 1, \\ 2, & x = 1 \end{array} \right .$$
is upper semicontinuous at $x = 1$. Does that mean that upper (lower) semicontinuity relaxes the requirement that a jump may occur up (down) at that point?

1. What is the story behind it?
2. Is there any real life example or analogy that we can relate to of this concept? (How to explain it to a precocious eight year old child?)
3. Why do we need the notion of upper (lower) semicontinuity?
4. What differentiate the notion of upper (lower) semicontinuity from that of left/right limit?
4. In what context is the notion of upper (lower) semicontinuity used widely?

 

[jgens]

I am not particularly well-versed with semi-continuous functions, but I think I can provide answers to some of your questions:

Does that mean that upper (lower) semicontinuity relaxes the requirement that a jump may occur up (down) at that point?

Very roughly speaking, a function f is upper (resp. lower) semi-continuous if the values of f in some neighborhood around x₀ are close to f(x₀) or are less than (resp. greater than) f(x₀). I think this is backward from what you have there, but I always get confused with directionality on graphs.

We can verify that this description is correct as follows: Let X be a topological space, let R^* denote the extended real numbers and consider an upper semi-continuous function f:X → R^*. Take α = f(x₀)+ε and consider the open neighborhood in f^-1([-∞,α)) containing x₀. For every x in this neighborhood, we see f(x) is close f(x₀) or f(x) < f(x₀).

2. Is there any real life example or analogy that we can relate to of this concept? (How to explain it to a precocious eight year old child?)

I do not know of any 'real life' applications of semi-continuity, so I am not sure how you could best relate this to anyone else. My best guess is that you could talk about graphs in R². For example, some high school courses explain that continuous functions are the kinds of functions whose graph you can draw without lifting your pencil. Likewise, for semi-continuity you could explain how this condition relaxes the conditions imposed by continuity, but also restricts some of the behaviors of the functions. I think mentioning it relaxes the upward/downward step discontinuity thing might be a good way to do this.

3. Why do we need the notion of upper (lower) semicontinuity?

I do not know why (or even if) we need it, but it can make things nicer. For example, in terms of semi-continuous functions, Urysohn's Lemma essentially states the existence of a continuous function between between upper and lower semi-continuous functions. The proof of the theorem can also be helped by utilizing upper and lower semi-continuity. In particular, once the desired function f has been constructed, we can show that it is continuous by checking that it is both upper and lower semi-continuous.

4. What differentiate the notion of upper (lower) semicontinuity from that of left/right limit?

Most of us have only seen limits in the contexts of metric spaces while global upper and lower semi-continuity have nice definitions that work for an arbitrary topological space.

I think you can generalize the limit concept to arbitrary topological spaces, in which case you could define local upper and lower semi-continuity, but I am not familiar with the concepts. In any case, I think this answers the question.

 

[perwiradua]

Thank you for the answer, jgens. My feeling is that the concept can be found in topology. However the scarcity of examples in the literature obviously does not help to reinforce one's understanding.

Roughly speaking, a function is continuous if its graph can be drawn without lifting the pencil. This is the kind of analogy that I am looking for when learning abstract mathematical concepts. It answers the question what it is about and makes learning such concepts more meaningful.