1. ## Bounded variation

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2. Originally Posted by Amer

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Given $\displaystyle f:[a,b] \to \mathbb{R}$ and a partition $\displaystyle P=\{a=t_0 < t_1 < ... < t_n=b \}$ then the vatiatin of fo over P is $\displaystyle V(f,P)=\sum_{n=0}^{n}|f(t_i)-f(t_{i-1}|$

Notice that this is very similar to a Riemann sum.

Now if we take the supreemum of all partitions of [a,b] This is called the total variation if $\displaystyle \sup_{P}=V(f,p) < \infty$

3. it is clear now thanks