Question on Functions of Bounded Variation

Question

We say that $f$ is of bounded variation over $[a,b]$ or $f\in BV[a,b]$ if

$V_f[a,b] = \sup \sum_{k=1}^n |f(x_k) - f(x_{k-1})| < \infty, $ where the supremum is taken over all possible partitions of $[a,b]$

My question is:

If $f \in BV[a,b],$ show that $|f(x)|\leq |f(a)| + V_f[a,b] \ \ $ for all $x\in [a,b],$

so that $f$ is bounded on $I=[a,b]$.

score 4 · Accepted Answer · answered May 10 '14 at 21:40

4

answered May 10 '14 at 21:40

Vincenzo Tibullo

so then you would say $||f(x)| - |f(a)|| \geq |f(x)| - |f(a)| $ therefore $V_f[a,b] + |f(a)| \geq |f(x)|$ right? – user143612 May 10 '14 at 21:51
@user143612: right – Vincenzo Tibullo May 10 '14 at 22:05

1 Answers1