I want to know the definition of non-periodic bounded variation function. I know the definition for periodic function of bounded variation, which is,
Let $f:[a,b]\to \mathcal c$ and $P=\{a=x_{0},x_{1},....,x_{n}=b\}$ be any partition of $[a,b]$. Set $$V_{P}(f)=\sum_{k=0}^{n-1}|f(x_{k+1})-f(x_{k})|.$$ If $\displaystyle V(f)=\sup_{P}V_{P}(f)<+\infty$ then we say that $f$ is of bounded variation the number $V(f)$ is called the total variation of $f$.
