Re: LSC

On 13 Jan 2006 09:01:01 -0800, "RamT" <rambotrout@xxxxxxxxx> wrote:

>> Not sure what you mean by "valid". Does the expression make
>> sense? Yes. Is it actually true? Depends on f.
>
>In the following I will use the other equivalent definition of
>lower semicontinuity. What I mean is, if x* belongs to dom(f)
>and that f(x*) is finite then checking if there exists nonempty
>neighborhood U of x* for all e > 0 such that f(x) >= f(x*) - e
>for all x in U, is possible. This is because it is possible to
>evalute the expression f(x) >= f(x*) - e. But when f(x*)=oo,
>can we evaluate the expression? Also oo - e is meaningless.

You need to make up your mind what the question is - the
answer to this is not quite the same as the answer to the
question you asked originally!

The reason the answer is not the same is that the two definitions
are _not_ equivalent, in the present context. They're equivalent
for real-valued functions, but not for extended-real-valued
functions.

The definition in terms of sequences _is_ equivalent to a
suitable _rephrasing_ of the definition above. Ask yourself
what does "f(x*) - e" mean, really? The point is just that
it's a number less than f(x*). If you rephrase things that
way you get an equivalent definition:

For every a < f(x*) there is a neighborhood U of x* such
that f >= a in U.

(Detail that's not important here, but which one should
mention because it may cause trouble elsewhere: In this
context oo - e is not usually taken to be meaningless,
it would usually be defined to equal oo. The problem
is that oo - e is not less than oo.)

>


David C. Ullrich
.

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