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**Extra info for Bounded Variation and Around**

**Example text**

Proof. 54) for the unilateral limits of a function ???? at some point ????0 . 54) exist for all ????0 ∈ (????, ????), as well as the unilateral limits ????(????+) and ????(????−). Given a regular function ???? : [????, ????] → ℝ, we consider the “average function” ???? of ???? defined by²⁰ { 1 (????(????−) + ????(????+)) for ???? ∈ ????1 (????) , ????(????) := { 2 ????(????) otherwise . 55) so it suffices to show that ????(????) is countable. Let ????0 ∈ ????0 (????) ∩ (????, ????), and hence ????(????0 −) = ????(????0 +) ≠ ????(????0 ). Assume, for example, that ????(????0 −) = ????(????0 +) < ????(????0 ); then, ???? := 12 (????(????0 ) − ????(????0 +)) > 0.

Being strictly monotone, the function ???? admits an inverse ????−1 : ???? → [????, ????] on its range ???? := ????([????, ????]). So, we may define a map ???? : ???? → ℝ by ????(????) := ????(????−1 (????)) (???? ∈ ????) . 58) By construction, this map satisfies ???? = ????∘???? on [????, ????]; we claim that ???? is continuous on the closure ???? of ????. First, we show that ???? is continuous on ???? = ????([????, ????]). Thus, fix ????∗ = ????(????∗ ) ∈ ????. If ????∗ ∈ ????(????), it is easy to see that ????∗ is an isolated point of ????, and there is nothing to prove.

38) holds in the new norm. 39) ‖????‖∞ := sup |????(????)| . 31. 39). e. ⦀????????⦀???? ≤ ⦀????⦀???? ⦀????⦀???? (????, ???? ∈ ????) . 42) Moreover, in case ???? ????→ ????([????, ????]), both norms ‖ ⋅ ‖???? and ⦀ ⋅ ⦀???? are equivalent. Proof. 11). In fact, we may have ‖????‖L∞ < ∞, but ‖????‖∞ = ∞, if ???? is unbounded on a nullset. 42). If ???? ????→ ????([????, ????]) with sharp imbedding constant ????(????, ????), then ⦀????⦀???? = ‖????‖???? + ‖????‖∞ ≤ (1 + ????(????, ????))‖????‖???? , while the converse estimate ‖????‖???? ≤ ⦀????⦀???? is of course trivial. 43). The following is a useful method to generate from a given function space a whole sequence of new function spaces by considering derivatives.