1. f n ( x ) is continuous, bounded, positive and f n ( x...

Roland Waters

Roland Waters

Answered

2022-07-01

1. f n ( x ) is continuous, bounded, positive and f n ( x ) f n + 1 ( x )
2. f ( x ) = lim f n ( x )
Question: Is f ( x ) a lower semicontinuous function?

Answer & Explanation

Marlee Guerra

Marlee Guerra

Expert

2022-07-02Added 25 answers

Consider, ( f n ) C [ 0 , 1 ] defined by
f n ( x ) = x n
Then, f n f pointwise on [0,1]
where, f : [ 0 , 1 ] R defined by
f ( x ) = { 0 0 x < 1 1 x = 1
Here, the function f is not lower semi continuous.
m f ( 1 ) = lim δ 0   i n f { f ( x ) : x ( 1 δ , 1 + δ ) } = 0
Hence, m f ( 1 ) f ( 1 )
Hence, f is not lower semi continuous at 1

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