# 微积分网课代修|积分学代写Integral Calculus代考|MTH-200 Growth of a Function on a Set

• 单变量微积分
• 多变量微积分
• 傅里叶级数
• 黎曼积分
• ODE
• 微分学

## 微积分作业代写calclulus代考|Growth of a Function on a Set

The calculus is concerned to a great extent with the growth of a function either at a point or on a set. The derivative measures the growth of a function at a point. A different concept is needed for the growth of a function on a set.

For a monotonic function $F$, growth on an interval $[a, b]$ can be simply measured as
$$|F(b)-F(a)|,$$
the absolute value of the difference of the end value and the beginning value. For a nonmonotonic function we should be interested in the all the ups and downs, not just the beginning and end values.
A measurement of the sums
$$\sum_{i=1}^{n}\left|F\left(b_{i}\right)-F\left(a_{i}\right)\right|$$
taken over nonoverlapping subintervals would be appropriate. This notion appears in the early literature and was formalized by Jordan in the late 19 th century under the terminology “variation of a function.”

For a first theoretical course in calculus we do not need (at first) the actual measurement of growth. What we do need is the notion that a function has zero growth or fails to grow on a set.
This leads to the following notions, explored in this chapter:

• A function $F$ does not grow on a set $E$.
• A set $N$ is a null set if the identity function $F(x)=x$ does not grow on N.
• A function $F$ is continuous at a point $x_{0}$ if $F$ does not grow on the singleton set $\left{x_{0}\right}$.
• A function $F$ is continuous if $F$ does not grow on any singleton set $\left{x_{0}\right}$.
• A function is absolutely continuous ${ }^{1}$ if it does not grow on any null set.
• A function $F$ with a zero derivative $F^{\prime}(x)=0$ at every point of a set $E$ does not grow on $E$.

The first five of these are definitions. The last one reveals the connection between zero derivatives and zero growth ${ }^{2}$.

## 微积分作业代写calclulus代考|Growth of a function on a set

The “change” of a function $F$ on an interval $[a, b]$ is given by the increment
$$\Delta F([a, b])=F(b)-F(a) .$$
What we wish to express is the variability of the function on a set $E$.
DEFINITION 3.1. A function $F: \mathbb{R} \rightarrow \mathbb{R}$ is said not to grow on a set $E$ if for every $\epsilon>0$ there can be found a full cover $\beta$ of that set $E$ so that
$$\sum_{i=1}^{n}\left|\Delta F\left(\left[a_{i}, b_{i}\right]\right)\right|<\epsilon$$
whenever the subpartition
$$\gamma=\left{\left(\left[a_{i}, b_{i}\right], x_{i}\right): i=1,2, \ldots, n\right}$$
is chosen from $\beta$.
Recall that in order for the subset $\gamma$ to be a subpartition, we require merely that the intervals $\left{\left[a_{i}, b_{i}\right]\right}$ do not overlap. The collection $\gamma$ here is not necessarily a partition (although it may be) and so we cannot use the letter ” $\pi . “$ It is what we have called a subpartition since it could be (but won’t be) expanded to be a partition.

## 微积分作业代写calclulus代考| Growth of a Function on a Set

$$|F(b)-F(a)|,$$

$$\sum_{i=1}^{n}\left|F\left(b_{i}\right)-F\left(a_{i}\right)\right|$$

## 微积分作业代写calclulus代考| Growth of a function on a set

$$\Delta F([a, b])=F(b)-F(a) .$$

$$\sum_{i=1}^{n}\left|\Delta F\left(\left[a_{i}, b_{i}\right]\right)\right|<\epsilon$$