(i) Show that every function f: 𝐑 →𝐑 of bounded variation is bounded, and that the limits limx →

Question

(i) Show that every function $f: \mathbf{R} \rightarrow \mathbf{R}$ of bounded variation is bounded, and that the limits $\lim _{x \rightarrow+\infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$, are well-defined. (ii) Give a counterexample of a bounded, continuous, compactly supported function $f$ that is not of bounded variation.

   (i) Show that every function $f: \mathbf{R} \rightarrow \mathbf{R}$ of bounded variation is bounded, and that the limits $\lim _{x \rightarrow+\infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$, are well-defined.
(ii) Give a counterexample of a bounded, continuous, compactly supported function $f$ that is not of bounded variation.

An Introduction To Measure Theory (January 2011 Draft)

Terence Tao 1st Edition

Chapter 1, Problem 37

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(i) Show that every function $f: \mathbf{R} \rightarrow \mathbf{R}$ of bounded variation is bounded, and that the limits $\lim _{x \rightarrow+\infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$, are well-defined. (ii) Give a counterexample of a bounded, continuous, compactly supported function $f$ that is not of bounded variation.

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