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 x→x0 f (x) ≤f (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 x∈B0(x0;δ)∩D f (x). Thus, liminf x→x0 f (x) =sup δ>0 h(δ) ≥h(δ0) ≥f (x0)−ε. Since ε is arbitrary, we obtain liminfx→x0 f (x) ≥f (x0). We now prove the converse. Suppose liminf x→x0 f (x) =sup δ>0 h(δ) ≥f (x0) and let ε >0. Since sup δ>0 h(δ) >f (x0)−ε, 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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