Theorem: Suppose f is bounded on [a,b], f has only finitely many points of discontinuity on [a,b], and α is continuous at every point at which f is discontinuous. Then f is in R(α).(adsbygoogle = window.adsbygoogle || []).push({});

Proof: Let ε>0 be given. Put M=sup|f(x)|, let E be the set of points at which f is discontinuous. Since E is finite and α is continuous at every point of E, we can cover E by finitely many disjoint intervals [u_{j}, v_{j}] in [a,b] such that the sum of the corresponding differences α(v_{j})-α(u_{j}) is less than ε.

Why this is true? I understand that we can cover E by those finitely many disjoint intervals, but I don't understand why we could cover E in such a way the sum of the corresponding differences α(v_{j})-α(u_{j}) would be less than epsilon.

Any help would be appreciated.

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# Would someone clarify this for me please? (from Rudin's analysis)

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