131 Exercises 5.3.1 Let D ⊂ R, f : D → R, and x0 be a limit point of D. Prove that liminfx →x0 f (x) ≤ limsupx →x0 f (x). 5.3.2 ▷ Find each of the following limits: 1 (a) limsupx →0 sin . x 1 (b) liminfx →0 sin . x cosx (c) limsupx →0 . x cosx (d) liminfx →0 . x 5.4 Lower Semicontinuity and Upper Semicontinuity The concept of semicontinuity is convenient for the study of maxima and minima of some discontinuous functions. Defnition 5.4.1 Let f : D → R and let x0 ∈ D. We say that f is lower semicontinuous (l.s.c.) at x0 if for every ε > 0, there exists δ > 0 such that f (x0) − ε < f (x) for all x ∈ B(x0;δ ) ∩ D. (5.4) Similarly, we say that f is upper semicontinuous (u.s.c.) at x0 if for every ε > 0, there exists δ > 0 such that f (x) < f (x0)+ ε for all x ∈ B(x0;δ ) ∩ D. Figure 5.1: Lower semicontinuity. It is clear that f is continuous at x0 if and only if f is lower semicontinuous and upper semicontinuous at this point. Theorem 5.4.1 Let f : D → R and let x0 ∈ D be a limit point of D. Then f is lower semicontinuous at x0 if and only if liminf f (x) ≥ f (x0). x→x0
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