Introduction to Mathematical Analysis I - Second Edition

94 3.7 LOWER SEMICONTINUITY AND UPPER SEMICONTINUITY Proof: Suppose lim x → ¯ x f ( x ) = `. Then for every ε > 0, there exists δ > 0 such that ` − ε < f ( x ) < ` + ε for all x ∈ B 0 ( ¯ x ; δ ) ∩ D . Since this also holds for every 0 < δ 0 < δ , we get ` − ε < g ( δ 0 ) ≤ ` + ε . It follows that ` − ε ≤ inf δ 0 > 0 g ( δ 0 ) ≤ ` + ε . Therefore, limsup x → ¯ x f ( x ) = ` since ε is arbitrary. The proof for the limit inferior is similar. The converse follows directly from (1) of Theorem 3.6.1 and Theorem 3.6.6 . Exercises 3.6.1 Let D ⊂ R , f : D → R , and ¯ x be a limit point of D . Prove that liminf x → ¯ x f ( x ) ≤ limsup x → ¯ x f ( x ) . 3.6.2 Find each of the following limits: (a) limsup x → 0 sin 1 x . (b) liminf x → 0 sin 1 x . (c) limsup x → 0 cos x x . (d) liminf x → 0 cos x x . 3.7 LOWER SEMICONTINUITY AND UPPER SEMICONTINUITY The concept of semicontinuity is convenient for the study of maxima and minima of some discontinuous functions. Definition 3.7.1 Let f : D → R and let ¯ x ∈ D . We say that f is lower semicontinuous (l.s.c.) at ¯ x if for every ε > 0, there exists δ > 0 such that f ( ¯ x ) − ε < f ( x ) for all x ∈ B ( ¯ x ; δ ) ∩ D . (3.12) Similarly, we say that f is upper semicontinuous (u.s.c.) at ¯ x if for every ε > 0, there exists δ > 0 such that f ( x ) < f ( ¯ x )+ ε for all x ∈ B ( ¯ x ; δ ) ∩ D . It is clear that f is continuous at ¯ x if and only if f is lower semicontinuous and upper semicontin- uous at this point.