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Let $f_n: \mathbb{R} \to \mathbb{R}$ be defined as follows:

$f_n$ is even.

$f_n(0) = \frac{1}{2}$

$f_n(x) = 0$ if $0

$f_n(x) = 1$ if $x> 2/n$

$f_n$ is linear on $(\frac{1}{n}, \frac{2}{n})$, and continuous on $\mathbb{R} - \{0\}$

I think each $f_n$ is USC, but the pointwise liminf is not. How am I wrong?

2 Answers 2

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There’s nothing wrong with your example; it’s the theorem that’s wrong. What is true is that the pointwise infimum of upper semicontinuous functions is upper semicontinuous, as is the pointwise limit if the sequence of functions is non-increasing.

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Perhaps you are confusing the pointwise limit with the pointwise infimum? Unless I'm mistaken, the sequence of functions is pointwise increasing, hence the infimum is just $f_1(x)$.

Edit: Last sentence fixed as per comment. And, ideed, as Brian's aswer point out, the theorem speaks about the (pointwise) infimum of the sequence, not of the pointwise limit (or pointwise liminf).

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    @CameronBuie: You're right, thanks2012-06-22