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Let $f:[a,b] \to \mathbb R$ a function of class $C^1$ on the interval $[a,b]$. Prove that:

i) $f$ is a function of bounded variation.

ii) The equality $V_a^b f= \int_a^b |f'(x)|dx$ holds.

My attempt at a solution.

For point i) what I did was:

By hypothesis, $f'$ is a continuous function on $[a,b]$, this means $f'$ attains a maximum and a minimum, call them $M$ and $m$ respectively and call $A=max\{|M|,|m|\}$. Then, $|f'(x)|\leq A$ for every $x \in [a,b]$. Let $\pi$ be a partition of $[a,b]$ with $\pi=\{x_0,x_1,...,x_n\}$ and $\sum(\pi)=\sum_{k=1}^n |f(x_k)-f(x_{k-1})|$. By the mean value theorem, for each subinterval $[x_{k-1},x_k]$, there exists $c_k \in [x_{k-1},x_k]$ such that $f(x_k)-f(x_{k-1})=f'(c)(x_k-x_{k-1}) \implies |f(x_k)-f(x_{k-1})|=|f'(c)|(x_k-x_{k-1})\leq A(x_k-x_{k-1})$.

From here, it follows that given a partition $\pi$ of $[a,b]$, $\sum_{k=1}^n |f(x_k)-f(x_{k-1})| \leq \sum_{k=1}^n A(x_k-x_{k-1})=A\sum_{k=1}^n x_k-x_{k-1}=A(b-a)$. This proves $f$ is of bounded variation.

I have problems with part ii), I must prove $V_a^b(f)\leq \int_a^b |f'(x)|dx$ and $\int_a^b |f'(x)|dx\leq V_a^b(f)$ but I don't know how to do this.

user100106
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    Hint: Use $\sum |f(x_k)-f(x_{k-1})| = \sum |f'(c_k)|(x_k-x_{k-1})$ –  Dec 01 '13 at 22:33
  • @John: Why don't you write it up? – user99680 Dec 01 '13 at 23:05
  • @user99680: The OP might want to have a try or might just want a hint. –  Dec 01 '13 at 23:15
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    @John Thanks for the suggestion, I am still stuck, could you give me a little push without telling me the complete answer? – user100106 Dec 01 '13 at 23:53
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    @user100106: In that equation, the left hand side is related to the total variation of $f$, while the right hand side is the Riemann sum of $\int_a^b|f'(x)|dx$. –  Dec 02 '13 at 00:02
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    By definition, $V_a^b(f)=sup {\sum_{k=1}^{n_p} |f(x_k)-f(x_{k-1})| P={x_0,...,x_{n_p}} }$, but I don't see so clearly that $\int_a^b |f'(x)|dx=sup{\sum_{k=1}^{n_p} |f'(c_k)||(x_k-x_{k-1}), c_k \in [x_k,x_{k-1}]}$ (the supremum of both sets runs through all posible partitions of $[a,b]$. – user100106 Dec 02 '13 at 00:44

1 Answers1

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Hint:

From the definition,

$$V_a^b(f) = \sup_P \big( \sum_{i=1}^{k_p} |f(x_i) - f(x_{i-1})|\big)\ ,$$

thus you can find a sequence of partition $P_n$ such that $||P_n|| \to 0$ and

$$V_a^b(f) = \lim_{n\to \infty} \big( \sum_{i=1}^{k_{p_n}} |f(x_i) - f(x_{i-1})|\big)\ \ \ \ (How?)$$

On the other hand, what is

$$\lim_{n\to \infty} \sum_{i=1}^{k_{p_n}} |f'(c_i)| (x_i -x_{i-1})\ \ ?$$

Why the limit exists?