131 Exercises 5.3.1 Let D⊂R, f : D→R, andx0 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: (a) limsupx →0 sin 1 x . (b) liminfx→0sin 1 x . (c) limsupx →0 cosx x . (d) liminfx→0 cosx 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. Definition 5.4.1 Let f : D→Rand 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 everyε >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→Rand let x0 ∈Dbe a limit point of D. Then f is lower semicontinuous at x0 if and only if liminf x→x0 f (x) ≥f (x0).
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