Let $f:\mathbb{R}^{+}\to \mathbb{R}$, and $\pi=(t_{0},\dots,t_{N})$. We define the variation of $f$ on the subdivision $\pi$, $V(f,\pi)$, as $$V(f,\pi)=\sum_{i=0}^{N-1}|f(t_{i+1})-f(t_{i})|$$or the variation of $f$ over $[a,b]$ as $$V(\left.f\right|_{[a,b]})=\sup_{\pi\in\Pi([a,b])}V(f,\pi)$$

Let $0\le a<b$, and let $f:\mathbb{R}^{+}\to \mathbb{R}$, $f$ is said to have bounded on $[a,b]$ if $$\sup_{\pi\in\Pi([a,b])}V(f,\pi)<\infty$$where $$\Pi([a,b])=\{ (t_{0},\dots,t_{N})\in\mathbb{R}^{N+1}:N\in\mathbb{N}^{*}\text{ and }a=t_{0}<t_{1}<\dots<t_{N}=b \}$$for $\pi=(t_{0},\dots,t_{N})$.

Let $f:\mathbb{R}^{+}\to \mathbb{R}$. $f$ is said to have finite variation if $\left.f\right|_{[0,t]}$ is of .