连续（Continuity） - 有界（Bounded）

连续（Continuity）

所有点连续 -> 一致连续 (uniform continuity) -> 绝对连续 -> 李普希兹连续（Lipschitz）

弱 ----> 强

【uniform continutity】

In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, where the maximum distance between f(x) and f(y) may depend on x and y themselves.

e.g. $1/x$ is not uniformly continous.

https://en.wikipedia.org/wiki/Uniform_continuity

【absolute continutiy】

绝对值也是一致连续的

https://en.wikipedia.org/wiki/Absolute_continuity

【Lipschitz Continuity】

Definition: 函数图像的曲线上任意两点连线的斜率"一致有界"，即任意两点的斜率都小于同一个常数，这个常数就是Lipschitz常数。

理解：

从局部看，我们可以取两个充分接近的点，如果这个时候斜率的极限存在的话，这个斜率的极限就是这个点的导数。也就是说函数可导，又是Lipschitz连续，那么导数有界。反过来，如果可导函数，导数有界，可以推出函数Lipschitz连续。

从整体看，Lipschitz连续要求函数在无限的区间上不能有超过线性的增长，所以x², e^x这些函数在无限区间上不是Lipschitz连续的。

Lipschitz连续的函数是比连续函数较更加“光滑”，但不一定是处处光滑的，比如|x|.但不光滑的点不多，放在一起是一个零测集，所以他是几乎处处的光滑的。

简单来说，Lipschitz连续就类似，一块地不仅没有河流什么的玩意儿阻隔，而且这块地上没有特别陡的坡。其中最陡的地方有多陡呢？这就是所谓的李普希兹常数

参考：

https://en.wikipedia.org/wiki/Lipschitz_continuity

有界（Bounded）

bounded -> Uniform boundedness

the sequence of functions $\{ f_n | f_n(x) = sin(nx) \}$ is uniformly bounded

the sequence of functions $\{ g_n | g_n(x) = nsin(x) \}$ is not uniformly bounded

【参考】

https://en.wikipedia.org/wiki/Uniform_boundedness

收敛（Convergence）

逐点收敛(pointwise convergence) -> 一致收敛(uniform convergence)

【pointwise convergence】

The sequence $f_n(x)$ converges pointwise to the function $f$, iff

for every $x$, $\lim_{x \to +\infty} f_n=f(x)$

【uniform convergence】

the sequence functions ${ S_n(x) }$ is uniformly convergent: if for every $\epsilon>0$, there exists a number N, such that for all $n>N$, $|f_n(x)-f(x)|<\epsilon$

https://en.wikipedia.org/wiki/Uniform_convergence

随机变量的收敛

研究一列随机变量是否会收敛到某个极限随机变量

https://en.wikipedia.org/wiki/Convergence_of_random_variables

【convergence in distribution】

the weakest form of convergence
related to central limit theorem

Definition:

A sequence $X_1$, $X_2$, ... of random variables is said to converge in distribution to a random variable X if

$\lim\limits_{n \to \infty} F_n(x)=F(x)$ for every $x\in\mathbb{R}$ at which $F$ is continues. (仅仅考虑$F(x)$连续的地方的分布函数值)

$X_n \overset{d}{\to} X$

【Convergence in probability】

related to the weak law of large numbers
related to the consistent estimator

meanning：the probability of an “unusual” outcome becomes smaller and smaller as the sequence progresses.

Definition: A sequence $\{X_n\}$ of random variables converges in probability towards the random variable $X$ if for all $\epsilon > 0$, $\lim\limits_{n \to \infty}{Pr(|X_n-X|>\epsilon)}=0$

$X_n \overset{p}{\to} X$

【Almost sure convergence】

类似于函数列收敛中pointwise convergence，

Definition：To say that the sequence X_n converges almost surely or almost everywhere or with probability 1 or strongly towards X means that,

$Pr(\lim\limits_{n \to \infty}{X_n=X})=1$

$X_n \overset{a.s.}{\to} X$

巴特西

连续（Continuity） - 有界（Bounded） - 收敛（Convergence）

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