135 It follows that f (x0) − ε < f (x) for all x ∈ B(x0;δ ) ∩ D. Therefore, f is lower semicontinuous. The proof for the upper semicontinuous case is similar. □ For every a ∈ R, we also defne La( f )= {x ∈ D : f (x) < a} = f − 1((−∞,a)) and Ua( f )= {x ∈ D : f (x) > a} = f − 1((a,∞)). Corollary 5.4.6 Let f : D → R. Then f is lower semicontinuous if and only if Ua( f ) is open in D for every a ∈ R. Similarly, f is upper semicontinuous if and only if La( f ) is open in D for every a ∈ R. Theorem 5.4.7 Let f : D → R. Then f is continuous if and only if for every a,b ∈ R with a < b, the set Oa,b = {x ∈ D : a < f (x) < b} = f − 1((a,b)) is an open set in D. Proof: Suppose f is continuous. Then f is lower semicontinuous and upper semicontinuous. Fix a,b ∈ R with a < b. Then Oa,b = Lb ∩Ua. By Theorem 5.4.6, the set Oa,b is open since it is the intersection of two open sets La and Ub. Let us prove the converse. We will only show that f is lower semicontinuous since the proof of upper semicontinuity is similar. For every a ∈ R, we have Ua( f )= {x ∈ D : f (x) > a} = [ f − 1((a,a + n)) n∈N Thus, Ua( f ) is open in D as it is a union of open sets in D. Therefore, f is lower semicontinuous by Corollary 5.4.6. □ Exercises 5.4.1 Let f be the function given by ( x2 , if x ̸ = 0; f (x)= −1, if x = 0. Prove that f is lower semicontinuous. 5.4.2 Let f be the function given by ( x2 , if x ̸ = 0; f (x)= 1, if x = 0. Prove that f is upper semicontinuous.
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