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# 数学代写|实分析代写Real Analysis代考|MATH351 Consequences of Continuity

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## 数学代写|实分析代写Real Analysis代考|Consequences of Continuity

Continuity is a powerful concept. There is much that we can conclude about any continuous function. In this section, we shall pursue some of these consequences and investigate the even richer rewards that accrue when differentiability is also brought in to play.

Theorem 3.6 (Continuous $\Rightarrow$ Bounded). If $f$ is continuous on the interval $[a, b]$, then there exist finite bounds $A$ and $B$ such that
$$A \leq f(x) \leq B$$
for all $x \in[a, b]$
Before proving this theorem, we note that we really do need all of the conditions. If $f$ only satisfies the intermediate value property, then it could be the function defined by
$$f(x)=x^{-1} \sin (1 / x), \quad x \neq 0 ; \quad f(0)=0$$
which is not bounded on $[0,1]$. If the endpoints of the interval are not included, then we could have a continuous function such as $f(x)=1 / x$ which is not bounded on $(0,1)$.

## 数学代写|实分析代写Real Analysis代考|Least Upper and Greatest Lower Bounds

If we want to patch up Cauchy’s first proof of the mean value theorem by assuming that the derivative is continuous on $[a, b]$, it is not enough to prove that a continuous function on a closed interval is bounded, it must actually achieve the best possible bounds. That is to say, if $f$ is continuous on $[a, b]$ then we must be able to find $c_1$ and $c_2$ in $[a, b]$ for which
$$f\left(c_1\right) \leq f(x) \leq f\left(c_2\right)$$
for all $x \in[a, b]$. The theorem we have just proved only promises us that bounds exist. It says nothing about how close these bounds come to the actual values of the function.
What we are usually interested in are the best possible bounds. In the case of $f(x)=x^3$ on $[-2,3]$, these are -8 and 27. Respectively, these are called the greatest lower bound and the least upper bound. The greatest lower bound is a lower bound with the property that any larger number is not a lower bound. Similarly, the least upper bound is an upper bound with the property that any smaller number is not an upper bound. Before we can ask whether or not $f$ achieves these best possible bounds, we must know whether they always exist. The precise definition is similar to the Archimedean understanding of a limit.

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## 数学代写|实分析代写Real Analysis代考|Consequences of Continuity

$$A \leq f(x) \leq B$$

$$f(x)=x^{-1} \sin (1 / x), \quad x \neq 0 ; \quad f(0)=0$$

## 数学代写|实分析代写Real Analysis代考|Least Upper and Greatest Lower Bounds

$$f\left(c_1\right) \leq f(x) \leq f\left(c_2\right)$$

## MATLAB代写

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