132 5.4 Lower Semicontinuity and Upper Semicontinuity Figure 5.2: Upper semicontinuity. Similarly, f is upper semicontinuous at x0 if and only if limsup f (x) ≤ f (x0). x→x0 Proof: Suppose f is lower semicontinuous at x0. Let ε > 0. Then there exists δ0 > 0 such that f (x0) − ε < f (x) for all x ∈ B(x0;δ0) ∩ D. This implies f (x0) − ε ≤ h(δ0), where h(δ )= inf f (x). x∈B0(x0;δ )∩D Thus, liminf f (x)= suph(δ ) ≥ h(δ0) ≥ f (x0) − ε. x→x0 δ >0 Since ε is arbitrary, we obtain liminfx →x0 f (x) ≥ f (x0). We now prove the converse. Suppose liminf f (x)= suph(δ ) ≥ f (x0) x→x0 δ >0 and let ε > 0. Since suph(δ ) > f (x0) − ε, δ >0 there exists δ > 0 such that h(δ ) > f (x0) − ε. This implies f (x) > f (x0) − ε for all x ∈ B0(x0;δ ) ∩ D. Since this is also true for x = x0, the function f is lower semicontinuous at x0. The proof for the upper semicontinuous case is similar. □
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