Introduction to Mathematical Analysis I - Second Edition

98 3.7 LOWER SEMICONTINUITY AND UPPER SEMICONTINUITY Proof: Suppose f is lower semicontinuous. Using Corollary 2.6.10 , we will prove that for every sequence { x k } in L a ( f ) that converges to a point ¯ x ∈ D , we get ¯ x ∈ L a ( f ) . For every k , since x k ∈ L a ( f ) , f ( x k ) ≤ a . Since f is lower semicontinuous at ¯ x , f ( ¯ x ) ≤ liminf k → ∞ f ( x k ) ≤ a . Thus, ¯ x ∈ L a ( f ) . It follows that L a ( f ) is closed. We now prove the converse. Fix any ¯ x ∈ D and ε > 0. Then the set G = { x ∈ D : f ( x ) > f ( ¯ x ) − ε } = D \ L f ( ¯ x ) − ε ( f ) is open in D and ¯ x ∈ G . Thus, there exists δ > 0 such that B ( ¯ x ; δ ) ∩ D ⊂ G . It follows that f ( ¯ x ) − ε < f ( x ) for all x ∈ B ( ¯ x ; δ ) ∩ D . Therefore, f is lower semicontinuous. The proof for the upper semicontinuous case is similar. For every a ∈ R , we also define L a ( f ) = { x ∈ D : f ( x ) < a } and U a ( f ) = { x ∈ D : f ( x ) > a } . Corollary 3.7.6 Let f : D → R . Then f is lower semicontinuous if and only if U a ( f ) is open in D for every a ∈ R . Similarly, f is upper semicontinuous if and only if L a ( f ) is open in D for every a ∈ R . Theorem 3.7.7 Let f : D → R . Then f is continuous if and only if for every a , b ∈ R with a < b , the set O a , b = { x ∈ D : a < f ( x ) < b } = f − 1 (( a , b )) is an open set in D . Proof: Suppose f is continuous. Then f is lower semicontinuous and upper semicontinuos. Fix a , b ∈ R with a < b . Then O a , b = L b ∩ U a . By Theorem 3.7.6 , the set O a , b is open since it is the intersection of two open sets L a and U b . Let us prove the converse. We will only show that f is lower semicontinuous since the proof of upper semicontinuity is similar. For every a ∈ R , we have U a ( f ) = { x ∈ D : f ( x ) > a } = ∪ n ∈ N f − 1 (( a , a + n )) Thus, U a ( f ) is open in D as it is a union of open sets in D . Therefore, f is lower semicontinuous by Corollary 3.7.6 .