Folland Chapter 3: Bounded Variation Difficulty Understanding
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Folland Proves in Lemma 3.26 that if $F in BV$ , then $T_F+F$ and $T_F-F$ are increasing, where: I can sort of follow the proof in Folland, but I don't understand why $T_F+F$ and $T_F-F$ are increasing for a BV function visually or graphically. Say $sin(x)$ on $[0,2 pi]$ is a BV function. I know that $T_F$ is increasing, but why are $T_F +sin(x)$ and $T_F-sin(x)$ increasing? I can't visualize why such is true. A picture would really help me understand. Thanks. I just need more visual help for understanding the definition of total variation.
real-analysis probability-theory measure-theory
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