# Thread: bounded variation

1. ## bounded variation

Let $f$ be a funtion from $R^+$ to $R$ such that $f$ is of bounded variation on $[0,n]$ where $n \in Z^+$. Let $V_f (0,n)$ be the total variation of $f$ on $[0,n]$. In addition, $f$ satisfies $\mid f(n)-f(0)\mid \leq M_n$ and $\sum_{n=1}^\infty M_n \leq \infty$.
Determine whether or not $lim_{n\rightarrow\infty} V_f (0,n)$ exists.

2. Originally Posted by Kat-M
Let $f$ be a funtion from $R^+$ to $R$ such that $f$ is of bounded variation on $[0,n]$ where $n \in Z^+$. Let $V_f (0,n)$ be the total variation of $f$ on $[0,n]$. In addition, $f$ satisfies $\mid f(n)-f(0)\mid \leq M_n$ and $\sum_{n=1}^\infty M_n \leq \infty$.
Determine whether or not $lim_{n\rightarrow\infty} V_f (0,n)$ exists.
the limit is not necessarily finite. for example $f(x)=\sin(\pi x)$ satisfies all the conditions in your problem but $V_f(0,n)=\int_0^n |f'(x)| \ dx = \int_0^{n \pi} |\cos x| \ dx=2n.$