# 微积分作业代写calclulus代考| Real-valued functions

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## 微积分作业代写calclulus代考|Basic concepts and definitions

In Chapters 2,3 , and 4 , we focus attention almost exclusively on scalarvalued functions of many variables, while in Chapter 5 we extend the ideas to vector-valued functions. In both contexts the following introduction to fundamental properties of multi-valued functions is invaluable. To start, we introduce some more notation and a pictorial view of what functions do.
In single-variable calculus we have the following scenario:
Let $y=f(x)$. The “graph” of $f$ is the set of ordered pairs ${(x, f(x))} \in \mathbb{R}^{2}$. This is shown graphically in Figure $1.11$ where the independent variable $x$ and dependent variable $y$ are plotted on mutually orthogonal axes.

## 微积分作业代写calclulus代考|Limits and continuity

In the next chapter we introduce and explore the concept of partial differentiation. In the lead up to that discussion it will be necessary to explain a number of concepts we shall then take for granted. Most importantly there is the notion of function continuity. For multivariable functions this will be
1.C Real-valued functions
21
discussed in detail in Section 2.B, but we can set the stage here with a short review of the subject as it relates to functions of one variable.

Function continuity is defined in terms of limiting processes. Mention has already been made of limit points of closed sets. We said that a point $\boldsymbol{a}$ is a limit point if any open sphere centred on $\boldsymbol{a}$, no matter how small in radius, contains points other than $\boldsymbol{a}$.

Similarly, segments of the real line possess the property that any open interval $I$, no matter how small, centred on a point $a$, contain points $x$ in $I$ different from $a$. The real line and any of its finite segments are therefore said to be complete: containing no gaps. This conjures up the notion of a set continuum, moving smoothly from one real value to another, never meeting any holes.

This notion gives critical meaning to the formalism x → a as the process of approaching a real value a along the real line. To be even more precise, we specify x → a− and x → a+ as meaning the respective approaches to a along the real line from “below” a (xa).

Now with thought given to single-variable functions defined on a domain
Df ⊂ R, the different approaches x → a− and x → a+ for a, x ∈ Df can have
all manner of implications for the function. Assuming a, x ∈ Df we define the
process of taking a limit of a function, which we denote either by

$\lim {x \rightarrow a^{-}} f(x), \lim {x \rightarrow a^{+}} f(x)$, or $\lim _{x \rightarrow a} f(x)$

as considering the sequence of values $f$ progressively takes as $x \rightarrow a^{-}$, $x \rightarrow a^{+}$, or in their combination. These considerations are of course separate to the question of what value $f$ actually takes at $a$. To summarize all of these ideas we have the following definition.

## 微积分作业代写calclulus代考|Limits and continuity

1.C 实值函数
21

Df ⊂ R 上的单变量函数，对于 a, x ∈ Df 的不同方法 x → a− 和 x → a+ 可以