Introduction to Mathematical Analysis I - Second Edition

95 Figure 3.6: Lower semicontinuity. Figure 3.7: Upper semicontinuity. Theorem 3.7.1 Let f : D → R and let ¯ x ∈ D be a limit point of D . Then f is lower semicontinuous at ¯ x if and only if liminf x → ¯ x f ( x ) ≥ f ( ¯ x ) . Similarly, f is upper semicontinuous at ¯ x if and only if limsup x → ¯ x f ( x ) ≤ f ( ¯ x ) . Proof: Suppose f is lower semicontinuous at ¯ x . Let ε > 0. Then there exists δ 0 > 0 such that f ( ¯ x ) − ε < f ( x ) for all x ∈ B ( ¯ x ; δ 0 ) ∩ D . This implies f ( ¯ x ) − ε ≤ h ( δ 0 ) , where h ( δ ) = inf x ∈ B 0 ( ¯ x ; δ ) ∩ D f ( x ) . Thus, liminf x → ¯ x f ( x ) = sup δ > 0 h ( δ ) ≥ h ( δ 0 ) ≥ f ( ¯ x ) − ε .