Let $f:[a,b]\to \mathbb{R}$ and let $D=\{x_o,x_1,...,x_n\}$ be a division of $[a,b]$. We say that $f$ is of bounded variation on $[a,b]$ if $\displaystyle \sup_{D\in \mathscr{D}} \sum_{i=1}^n |f(x_i)-f(x_{i-1})|<\infty$, where $\mathscr{D}$ is a collection of divisions of $[a,b]$.
My question is: Does Bounded Variation implies that a function is Bounded? Thanks.
Yes, because for arbitrary $x$ you can easily find $D$ showing that $|f(x)-f(a)|\le $ the given bound.