连续功能据/h1>
已经有一个帐户?据一种href="//www.parkandroid.com/account/login/?next=/wiki/continuous-functions/" class="ax-click" data-ax-id="clicked_signup_modal_login" data-ax-type="link">这里登录。据/a>
在据一种href="//www.parkandroid.com/wiki/calculus/" class="wiki_link" title="结石" target="_blank">结石据/a>, 一种据strong>连续函数据/strong>是一个真实值的函数,其图表没有任何休息或孔。连续性奠定了基础的基础据一种href="//www.parkandroid.com/wiki/intermediate-value-theorem/" class="wiki_link" title="中间值定理" target="_blank">中间值定理据/a>和据一种href="//www.parkandroid.com/wiki/extreme-value-theorem/" class="wiki_link" title="极值定理" target="_blank">极值定理据/a>.他们在某种意义上实分析的``最好的”功能成为可能,许多证据依赖于连续函数逼近任意函数。据/p>
内容据/h4>
定义据/h2>
在微积分中,知道函数是否连续是必要的,因为只有当函数是连续的时,才能进行微分。连续性的概念很简单:如果函数的图形在某一区间内没有任何间断或空洞,则该函数在该区间内是连续的。因此,简单地画出图形就可以告诉你函数是否是连续的。然而,并不是所有的函数都很容易画出来,有时我们需要使用连续性的定义来确定一个函数的连续性。连续函数的数学定义如下:据/p>
对于一个功能据span class="katex"> F据/span>(据/span>X据/span>)据/span>是在一个点的连续据span class="katex"> X据/span>=据/span>一种据/span>,它必须满足以下条件的前三个条件:据/p>
(一世)据/strong> F据/span>(据/span>一种据/span>)据/span>存在。据/p>
(ii)据/strong> X据/span>→据/span>一种据/span>林据/span>F据/span>(据/span>X据/span>)据/span>存在。据/p>
(iii)据/strong> X据/span>→据/span>一种据/span>林据/span>F据/span>(据/span>X据/span>)据/span>=据/span>F据/span>(据/span>一种据/span>)据/span>.据/span>
(iv)据/strong>对所有人据span class="katex"> ε据/span>>据/span>0.据/span>, 那里存在据span class="katex"> δ据/span>>据/span>0.据/span>这样据span class="katex"> |据/span>X据/span>-据/span>一种据/span>|据/span>据据/span>δ据/span>那据/span>X据/span>据/span>=据/span>一种据/span>暗示据span class="katex"> |据/span>|据/span>F据/span>(据/span>X据/span>)据/span>-据/span>F据/span>(据/span>一种据/span>)据/span>|据/span>|据/span>据据/span>ε据/span>.据/span>
当我们说一个函数据span class="katex"> F据/span>是持续的据span class="katex"> [据/span>一种据/span>那据/span>B.据/span>]据/span>那据/span>这意味着,对于间隔中的所有元素,满足上述条件。但请注意,在试图证明连续性时据span class="katex"> [据/span>一种据/span>那据/span>B.据/span>]据/span>那据/span>我们不需要考虑LHL据span class="katex"> 一种据/span>和rhl据span class="katex"> B.据/span>作为左边的点据span class="katex"> 一种据/span>和权据span class="katex"> B.据/span>不包括在据span class="katex"> [据/span>一种据/span>那据/span>B.据/span>]据/span>.据/span>我们只考虑RHL据span class="katex"> 一种据/span>和lhl for.据span class="katex"> B.据/span>.据/span>
例子据/h2>
以下是一些示例问题。使用上面的定义,尝试确定它们是否是连续的。据/p>
是函数据span class="katex"> F据/span>(据/span>X据/span>)据/span>=据/span>{据/span>2据/span>X据/span>+据/span>1据/span>(据/span>X据/span>据据/span>3.据/span>)据/span>3.据/span>X据/span>-据/span>2据/span>(据/span>X据/span>≥据/span>3.据/span>)据/span>全部持续据span class="katex"> X据/span>∈据/span>R.据/span>还是据/span>
我们知道的图形据span class="katex"> y据/span>=据/span>2据/span>X据/span>+据/span>1据/span>和据span class="katex"> y据/span>=据/span>3.据/span>X据/span>-据/span>2据/span>是连续的,所以我们只需要看看函数在据span class="katex"> X据/span>=据/span>3.据/span>.据/span>该过程简单地使用上面的定义,如下:据/p>
(一世)据/strong>自据span class="katex"> F据/span>(据/span>3.据/span>)据/span>=据/span>3.据/span>×据/span>3.据/span>-据/span>2据/span>=据/span>7.据/span>那据/span> F据/span>(据/span>3.据/span>)据/span>存在。据/p>
(ii)据/strong>为了看看是否存在限制,我们必须检查两侧的限制。左手和右手限制是据/p>
X据/span>→据/span>3.据/span>-据/span>林据/span>F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>→据/span>3.据/span>-据/span>林据/span>(据/span>2据/span>X据/span>+据/span>1据/span>)据/span>=据/span>2据/span>×据/span>3.据/span>+据/span>1据/span>=据/span>7.据/span>和据/span>X据/span>→据/span>3.据/span>+据/span>林据/span>F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>→据/span>3.据/span>+据/span>林据/span>(据/span>3.据/span>X据/span>-据/span>2据/span>)据/span>=据/span>3.据/span>×据/span>3.据/span>-据/span>2据/span>=据/span>7.据/span>那据/span>
分别。因为两侧的极限相等,据span class="katex"> X据/span>→据/span>3.据/span>林据/span>F据/span>(据/span>X据/span>)据/span>存在。据/p>
(iii)据/strong>现在来自(i)和(ii),我们有据span class="katex"> X据/span>→据/span>3.据/span>林据/span>F据/span>(据/span>X据/span>)据/span>=据/span>F据/span>(据/span>3.据/span>)据/span>=据/span>7.据/span>那据/span>所以这个功能是连续的据span class="katex"> X据/span>=据/span>3.据/span>.据/span> □据/span>
是函数据span class="katex"> F据/span>(据/span>X据/span>)据/span>=据/span>⎩据/span>⎪据/span>⎨据/span>⎪据/span>⎧据/span>X据/span>+据/span>1据/span>X据/span>2据/span>2据/span>X据/span>-据/span>1据/span>(据/span>X据/span>据据/span>2据/span>)据/span>(据/span>X据/span>=据/span>2据/span>)据/span>(据/span>X据/span>>据/span>2据/span>)据/span>全部持续据span class="katex"> X据/span>∈据/span>R.据/span>还是据/span>
我们知道的图形据span class="katex"> y据/span>=据/span>X据/span>+据/span>1据/span>那据/span> y据/span>=据/span>X据/span>2据/span>那据/span>和据span class="katex"> y据/span>=据/span>2据/span>X据/span>-据/span>1据/span>是连续的,所以我们只需要看看函数在据span class="katex"> X据/span>=据/span>2据/span>.据/span>再次,我们使用相同的程序,如下图所示:据/p>
(一世)据/strong>自据span class="katex"> F据/span>(据/span>2据/span>)据/span>=据/span>2据/span>2据/span>=据/span>4.据/span>那据/span> F据/span>(据/span>2据/span>)据/span>存在。据/p>
(ii)据/strong>左手和右手限制是据/p>
X据/span>→据/span>2据/span>-据/span>林据/span>F据/span>(据/span>X据/span>)据/span>X据/span>→据/span>2据/span>+据/span>林据/span>F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>→据/span>2据/span>-据/span>林据/span>(据/span>X据/span>+据/span>1据/span>)据/span>=据/span>3.据/span>=据/span>X据/span>→据/span>2据/span>+据/span>林据/span>(据/span>2据/span>X据/span>-据/span>1据/span>)据/span>=据/span>3.据/span>那据/span>
分别。所以,据span class="katex"> X据/span>→据/span>2据/span>-据/span>林据/span>F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>→据/span>2据/span>+据/span>林据/span>F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>→据/span>3.据/span>林据/span>F据/span>(据/span>X据/span>)据/span>=据/span>3.据/span>.据/span>
(iii)据/strong>现在来自(i)和(ii),我们有据span class="katex"> X据/span>→据/span>2据/span>林据/span>F据/span>(据/span>X据/span>)据/span>据/span>=据/span>F据/span>(据/span>2据/span>)据/span>那据/span>所以功能不是在连续据span class="katex"> X据/span>=据/span>2据/span>.据/span>
该函数的图形看起来就像上图中。注意,有一个“洞”,在据span class="katex"> X据/span>=据/span>2据/span>那据/span>这导致不连续性。据span class="katex"> □据/span>
是函数据span class="katex"> F据/span>(据/span>X据/span>)据/span>=据/span>{据/span>-据/span>X据/span>3.据/span>+据/span>X据/span>+据/span>1据/span>(据/span>X据/span>≤.据/span>1据/span>)据/span>2据/span>X据/span>2据/span>+据/span>3.据/span>X据/span>-据/span>2据/span>(据/span>X据/span>>据/span>1据/span>)据/span>全部持续据span class="katex"> X据/span>∈据/span>R.据/span>还是据/span>
我们知道的图形据span class="katex"> y据/span>=据/span>-据/span>X据/span>3.据/span>+据/span>X据/span>+据/span>1据/span>和据span class="katex"> y据/span>=据/span>2据/span>X据/span>2据/span>+据/span>3.据/span>X据/span>-据/span>2据/span>是连续的,所以我们只需要看看函数在据span class="katex"> X据/span>=据/span>1据/span>.据/span>
(一世)据/strong>自据span class="katex"> F据/span>(据/span>1据/span>)据/span>=据/span>1据/span>那据/span> F据/span>(据/span>1据/span>)据/span>存在。据/p>
(ii)据/strong>左手和右手限制是据/p>
X据/span>→据/span>1据/span>-据/span>林据/span>F据/span>(据/span>X据/span>)据/span>X据/span>→据/span>1据/span>+据/span>林据/span>F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>→据/span>1据/span>-据/span>林据/span>(据/span>-据/span>X据/span>3.据/span>+据/span>X据/span>+据/span>1据/span>)据/span>=据/span>1据/span>=据/span>X据/span>→据/span>1据/span>+据/span>林据/span>(据/span>2据/span>X据/span>2据/span>+据/span>3.据/span>X据/span>-据/span>2据/span>)据/span>=据/span>3.据/span>那据/span>
分别。由于左侧限制和右侧限制不等于,据span class="katex"> X据/span>→据/span>1据/span>林据/span>F据/span>(据/span>X据/span>)据/span>不存在,所以功能据span class="katex"> F据/span>(据/span>X据/span>)据/span>不连续据span class="katex"> X据/span>=据/span>1据/span>.据/span>
该函数的图形看起来就像上图中。注意,有一个“破发”的据span class="katex"> X据/span>=据/span>1据/span>那据/span>这导致不连续性。据span class="katex"> □据/span>
是函数据span class="katex"> F据/span>(据/span>X据/span>)据/span>=据/span>{据/span>-据/span>COS.据/span>X据/span>E.据/span>X据/span>-据/span>2据/span>(据/span>X据/span>据据/span>0.据/span>)据/span>(据/span>X据/span>≥据/span>0.据/span>)据/span>全部持续据span class="katex"> X据/span>∈据/span>R.据/span>还是据/span>
我们知道的图形据span class="katex"> y据/span>=据/span>-据/span>COS.据/span>X据/span>和据span class="katex"> y据/span>=据/span>E.据/span>X据/span>-据/span>2据/span>是连续的,所以我们只需要看看函数在据span class="katex"> X据/span>=据/span>0.据/span>.据/span>
(一世)据/strong>自据span class="katex"> F据/span>(据/span>0.据/span>)据/span>=据/span>E.据/span>0.据/span>-据/span>2据/span>=据/span>-据/span>1据/span>那据/span> F据/span>(据/span>0.据/span>)据/span>存在。据/p>
(ii)据/strong>左手和右手限制是据/p>
X据/span>→据/span>0.据/span>-据/span>林据/span>F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>→据/span>0.据/span>-据/span>林据/span>(据/span>-据/span>COS.据/span>X据/span>)据/span>X据/span>→据/span>0.据/span>+据/span>林据/span>F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>→据/span>0.据/span>+据/span>林据/span>(据/span>E.据/span>X据/span>-据/span>2据/span>)据/span>=据/span>-据/span>1据/span>=据/span>-据/span>1据/span>那据/span>
分别。所以,据span class="katex"> X据/span>→据/span>0.据/span>-据/span>林据/span>F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>→据/span>0.据/span>+据/span>林据/span>F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>→据/span>0.据/span>林据/span>F据/span>(据/span>X据/span>)据/span>=据/span>-据/span>1据/span>.据/span>
(iii)据/strong>现在来自(i)和(ii),我们有据span class="katex"> X据/span>→据/span>2据/span>林据/span>F据/span>(据/span>X据/span>)据/span>=据/span>F据/span>(据/span>2据/span>)据/span>=据/span>-据/span>1据/span>那据/span>所以这个功能是连续的据span class="katex"> X据/span>=据/span>0.据/span>.据/span> □据/span>
中间价值定理据/h2>
主要文章:据一种href="//www.parkandroid.com/wiki/intermediate-value-theorem/" class="wiki_link" title="中间价值定理" target="_blank">中间价值定理据/a>
极值定理据/h2>
主要文章:据一种href="//www.parkandroid.com/wiki/extreme-value-theorem/" class="wiki_link" title="极值定理" target="_blank">极值定理据/a>
一致连续性据/h2>
统一连续性的连续性更强的概念。请注意,在定义上的连续性的间隔据span class="katex">
一世据/span>那据/span>我们说, ”据span class="katex">
F据/span>必须为所有的连续据span class="katex">
X据/span>0.据/span>∈据/span>一世据/span>那据/span>“这意味着所有据span class="katex">
X据/span>0.据/span>∈据/span>一世据/span>而对于一些给定据span class="katex">
ε据/span>>据/span>0.据/span>我们必须能够挑选据span class="katex">
δ据/span>>据/span>0.据/span>这样据span class="katex">
|据/span>X据/span>-据/span>X据/span>0.据/span>|据/span>据据/span>δ据/span>暗示据span class="katex">
|据/span>|据/span>F据/span>(据/span>X据/span>)据/span>-据/span>F据/span>(据/span>X据/span>0.据/span>)据/span>|据/span>|据/span>据据/span>ε据/span>.然而,请注意据span class="katex">
X据/span>1据/span>那据/span>X据/span>2据/span>∈据/span>一世据/span>这据span class="katex">
δ据/span>X据/span>1据/span>我们挑选据span class="katex">
X据/span>=据/span>X据/span>1据/span>可以是来自不同据span class="katex">
δ据/span>X据/span>2据/span>我们挑选据span class="katex">
X据/span>=据/span>X据/span>2据/span>.统一的连续性允许我们选择一个据span class="katex">
δ据/span>对所有人据span class="katex">
X据/span>那据/span>y据/span>∈据/span>一世据/span>,这是什么使比的间隔连续性更强一致连续的概念。我们正式定义的一致连续性如下:据/p>
让据span class="katex">
一世据/span>⊂据/span>R.据/span>.一个功能据span class="katex">
F据/span>:据/span>一世据/span>→据/span>R.据/span>是均匀地连续据span class="katex">
一世据/span>如果据/p>
(一世)据/strong>对所有人据span class="katex">
ε据/span>>据/span>0.据/span>, 那里存在据span class="katex">
δ据/span>>据/span>0.据/span>这样,对于所有人据span class="katex">
X据/span>那据/span>y据/span>∈据/span>一世据/span>那据/span>|据/span>X据/span>-据/span>y据/span>|据/span>据据/span>δ据/span>暗示据span class="katex">
|据/span>|据/span>F据/span>(据/span>X据/span>)据/span>-据/span>F据/span>(据/span>y据/span>)据/span>|据/span>|据/span>据据/span>ε据/span>;据/span>
(ii)据/strong>
∀据/span>ε据/span>>据/span>0.据/span>那据/span>∃据/span>δ据/span>>据/span>0.据/span>那据/span>∀据/span>X据/span>那据/span>y据/span>∈据/span>一世据/span>那据/span>|据/span>X据/span>-据/span>y据/span>|据/span>据据/span>δ据/span>⟹据/span>|据/span>|据/span>F据/span>(据/span>X据/span>)据/span>-据/span>F据/span>(据/span>y据/span>)据/span>|据/span>|据/span>据据/span>ε据/span>.据/span>
换句话说,一个函数据span class="katex">
F据/span>是,如果一致连续据span class="katex">
δ据/span>独立于任何特定点选择。这种较强的连续性概念具有一些非常强大的结果,我们将进一步审查,但首先是一个例子。让我们展示这一点据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>2据/span>是均匀地连续据span class="katex">
[据/span>-据/span>2据/span>那据/span>3.据/span>]据/span>.据/p>
让据span class="katex">
ε据/span>>据/span>0.据/span>我们现在寻求一些据span class="katex">
δ据/span>>据/span>0.据/span>这样,对于所有人据span class="katex">
X据/span>那据/span>y据/span>∈据/span>[据/span>-据/span>2据/span>那据/span>3.据/span>]据/span>如果据span class="katex">
|据/span>X据/span>-据/span>y据/span>|据/span>据据/span>δ据/span>我们有据span class="katex">
|据/span>|据/span>F据/span>(据/span>X据/span>)据/span>-据/span>F据/span>(据/span>y据/span>)据/span>|据/span>据据/span>ε据/span>.请看下面的不等式指出我们对据span class="katex">
[据/span>-据/span>2据/span>那据/span>3.据/span>]据/span>:据/span>
|据/span>|据/span>F据/span>(据/span>X据/span>)据/span>-据/span>F据/span>(据/span>y据/span>)据/span>|据/span>|据/span>=据/span>|据/span>|据/span>X据/span>2据/span>-据/span>y据/span>2据/span>|据/span>|据/span>=据/span>|据/span>X据/span>-据/span>y据/span>|据/span>|据/span>X据/span>+据/span>y据/span>|据/span>≤.据/span>9.据/span>|据/span>X据/span>-据/span>y据/span>|据/span>.据/span>看起来是这样的选择据span class="katex">
δ据/span>=据/span>9.据/span>ε据/span>可能是个好主意!让我们来看看。定义据span class="katex">
δ据/span>=据/span>9.据/span>ε据/span>然后假设据span class="katex">
|据/span>X据/span>-据/span>y据/span>|据/span>据据/span>δ据/span>那据/span>我们有据span class="katex-display">
|据/span>|据/span>F据/span>(据/span>X据/span>)据/span>-据/span>F据/span>(据/span>y据/span>)据/span>|据/span>|据/span>=据/span>|据/span>|据/span>X据/span>2据/span>-据/span>y据/span>2据/span>|据/span>|据/span>=据/span>|据/span>X据/span>-据/span>y据/span>|据/span>|据/span>X据/span>+据/span>y据/span>|据/span>≤.据/span>9.据/span>|据/span>X据/span>-据/span>y据/span>|据/span>据据/span>9.据/span>δ据/span>=据/span>9.据/span>9.据/span>ε据/span>=据/span>ε据/span>.据/span>因此据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>2据/span>是均匀地连续据span class="katex">
[据/span>-据/span>2据/span>那据/span>3.据/span>]据/span>.据/span>□据/span>
乍一看,连续性和均匀连续性似乎相当相似。事实上,除了我们在选择时考虑的因素外,它们的定义似乎几乎是相同的据span class="katex">
δ据/span>;据/span>我们会看到这使得这成为一个不同的世界。下面我们分别有两个正式的连续性和统一连续性定义:据/p>
连续性据/strong>在据span class="katex">
一世据/span>:据/span> 对所有人据span class="katex">
ε据/span>>据/span>0.据/span>, 那里存在据span class="katex">
δ据/span>>据/span>0.据/span>,所有人都在哪里据span class="katex">
y据/span>∈据/span>一世据/span>那据/span>|据/span>X据/span>-据/span>y据/span>|据/span>据据/span>δ据/span>暗示据span class="katex">
|据/span>|据/span>F据/span>(据/span>X据/span>)据/span>-据/span>F据/span>(据/span>y据/span>)据/span>|据/span>|据/span>据据/span>ε据/span>.据/span>
一致连续性据/strong>在据span class="katex">
一世据/span>:据/span> 对所有人据span class="katex">
ε据/span>>据/span>0.据/span>, 那里存在据span class="katex">
δ据/span>>据/span>0.据/span>,所以对所有人来说据span class="katex">
X据/span>那据/span>y据/span>∈据/span>一世据/span>那据/span>|据/span>X据/span>-据/span>y据/span>|据/span>据据/span>δ据/span>暗示据span class="katex">
|据/span>|据/span>F据/span>(据/span>X据/span>)据/span>-据/span>F据/span>(据/span>y据/span>)据/span>|据/span>|据/span>据据/span>ε据/span>.据/span>
我们之前提到的,统一的连续性是一个比连续性更强的概念;我们现在证明,实际上统一的连续性意味着连续性。据/p>
如果据span class="katex">
F据/span>是均匀地连续据span class="katex">
一世据/span>⊂据/span>R.据/span>那据/span>然后据span class="katex">
F据/span>是持续的据span class="katex">
一世据/span>.据/p>
证明据/strong>: 假使,假设据span class="katex">
F据/span>是均匀地连续据span class="katex">
一世据/span>⊂据/span>R.据/span>那就是那样的据span class="katex">
一世据/span>我们都知道据span class="katex">
ε据/span>>据/span>0.据/span>, 那里存在据span class="katex">
δ据/span>>据/span>0.据/span>这样,对于所有人据span class="katex">
X据/span>那据/span>y据/span>∈据/span>一世据/span>那据/span>|据/span>X据/span>-据/span>y据/span>|据/span>据据/span>δ据/span>暗示据span class="katex">
|据/span>|据/span>F据/span>(据/span>X据/span>)据/span>-据/span>F据/span>(据/span>y据/span>)据/span>|据/span>|据/span>据据/span>ε据/span>.让据span class="katex">
X据/span>0.据/span>∈据/span>一世据/span>然后让据span class="katex">
ε据/span>>据/span>0.据/span>,那么我们现在寻求据span class="katex">
δ据/span>>据/span>0.据/span>这样据span class="katex">
|据/span>X据/span>-据/span>X据/span>0.据/span>|据/span>据据/span>δ据/span>暗示据span class="katex">
|据/span>|据/span>F据/span>(据/span>X据/span>)据/span>-据/span>F据/span>(据/span>X据/span>0.据/span>)据/span>|据/span>|据/span>据据/span>ε据/span>.通过假设据span class="katex">
F据/span>是均匀连续,因而存在据span class="katex">
δ据/span>>据/span>0.据/span>,所以对所有人来说据span class="katex">
X据/span>那据/span>y据/span>∈据/span>一世据/span>那据/span>|据/span>X据/span>-据/span>y据/span>|据/span>据据/span>δ据/span>暗示据span class="katex">
|据/span>|据/span>F据/span>(据/span>X据/span>)据/span>-据/span>F据/span>(据/span>y据/span>)据/span>|据/span>|据/span>据据/span>ε据/span>因此挑选了这一点据span class="katex">
δ据/span>确保据span class="katex">
|据/span>X据/span>-据/span>X据/span>0.据/span>|据/span>据据/span>δ据/span>暗示据span class="katex">
|据/span>|据/span>F据/span>(据/span>X据/span>)据/span>-据/span>F据/span>(据/span>X据/span>0.据/span>)据/span>|据/span>|据/span>据据/span>ε据/span>.因此,统一的连续性意味着连续性。据span class="katex">
□据/span>
我们现在考虑交谈。连续性意味着均匀的连续性吗?让我们看,假设连续性意味着统一的连续性。这意味着对于该功能据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>1据/span>哪个是连续的据span class="katex">
(据/span>0.据/span>那据/span>∞据/span>)据/span>那据/span>我们会有那个据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>1据/span>是均匀地连续据span class="katex">
(据/span>0.据/span>那据/span>∞据/span>)据/span>.让据span class="katex">
ε据/span>=据/span>1据/span>,并限定两个序列据span class="katex">
(据/span>X据/span>N据/span>)据/span>N据/span>=据/span>1据/span>∞据/span>那据/span>(据/span>y据/span>N据/span>)据/span>N据/span>=据/span>1据/span>∞据/span>⊂据/span>R.据/span>, 在哪里据span class="katex">
X据/span>N据/span>=据/span>N据/span>1据/span>和据span class="katex">
y据/span>N据/span>=据/span>N据/span>2据/span>1据/span>并注意据span class="katex">
N据/span>→据/span>∞据/span>林据/span>X据/span>N据/span>=据/span>N据/span>→据/span>∞据/span>林据/span>y据/span>N据/span>=据/span>0.据/span>.所以我们有据span class="katex-display">
|据/span>|据/span>F据/span>(据/span>X据/span>N据/span>)据/span>-据/span>F据/span>(据/span>y据/span>N据/span>)据/span>|据/span>|据/span>=据/span>|据/span>|据/span>|据/span>|据/span>N据/span>1据/span>1据/span>-据/span>N据/span>2据/span>1据/span>1据/span>|据/span>|据/span>|据/span>|据/span>=据/span>|据/span>|据/span>N据/span>-据/span>N据/span>2据/span>|据/span>|据/span>=据/span>|据/span>|据/span>N据/span>2据/span>-据/span>N据/span>|据/span>|据/span>.据/span>我们看到了据span class="katex">
N据/span>>据/span>2据/span>我们有据span class="katex">
|据/span>|据/span>F据/span>(据/span>X据/span>N据/span>)据/span>-据/span>F据/span>(据/span>y据/span>N据/span>)据/span>|据/span>|据/span>>据/span>1据/span>那据/span>这违背了我们的假设是据span class="katex">
F据/span>是均匀连续的。因此,连续性并不意味着一致连续性。据/p>
但是,并非所有的希望都丢失了。我们可以将一个条件添加到我们的连续功能据span class="katex">
F据/span>要使它一致连续,我们需要据span class="katex">
F据/span>在封闭和有界间隔上连续。据/p>
如果据span class="katex">
F据/span>是持续的据span class="katex">
[据/span>一种据/span>那据/span>B.据/span>]据/span>⊂据/span>R.据/span>那据/span>在哪里据span class="katex">
[据/span>一种据/span>那据/span>B.据/span>]据/span>是封闭和有界的,那么据span class="katex">
F据/span>是均匀地连续据span class="katex">
[据/span>一种据/span>那据/span>B.据/span>]据/span>.据/p>
证明据/strong>:我们假设了一个矛盾:据span class="katex">
F据/span>是持续的据span class="katex">
[据/span>一种据/span>那据/span>B.据/span>]据/span>⊂据/span>R.据/span>那据/span>在哪里据span class="katex">
[据/span>一种据/span>那据/span>B.据/span>]据/span>被关闭,界,和据span class="katex">
F据/span>是据strong>不是据/strong>均匀连续据span class="katex">
[据/span>一种据/span>那据/span>B.据/span>]据/span>,这意味着据span class="katex">