Continuity
All points in a row -> uniformly continuous (uniform continuity) -> absolutely continuous -> Lipschitz continuous ( Lipschitz)
Weak ----> strong
【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】
The absolute value is uniformly continuous
https://en.wikipedia.org/wiki/Absolute_continuity
【Lipschitz Continuity】
Definition: connecting any two points on the slope of the curve is a function of the image "uniformly bounded", i.e., the slope between any two points is less than with a constant, this constant is the Lipschitz constant.
understanding:
From the local perspective, we can take two sufficiently close to the point, if this time limit slope exists, limit the slope is the derivative of this point. That function may lead, is Lipschitz continuous, then the derivative bounded. Conversely, if differentiable function, the derivative bounded, Lipschitz continuous function can be introduced.
On the whole, a continuous function of the required Lipschitz not have increased more than in a linear infinite interval, so that X 2 , E X these functions are not infinite intervals Lipschitz continuous.
Lipschitz continuous function is more "smooth" than the relatively continuous function, but not necessarily always smooth, such as | x |. The point is not smooth, but not much, to put together a set of measure zero, so he is almost everywhere Smooth.
In simple terms, Lipschitz continuous on similar piece of land not only did what stuff blocking the river, and this land is not particularly steep slope. Which steepest how steep it? This is called the Lipschitz constant
reference:
https://en.wikipedia.org/wiki/Lipschitz_continuity
Bounded (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
【reference】
https://en.wikipedia.org/wiki/Uniform_boundedness
Convergence (Convergence)
Pointwise convergence (pointwise convergence) -> uniform convergence (uniform convergence)
【uniform convergence】
the sequence functions ${ S_n(x) }$ isuniformly convergent: if for every $\episilon>0$, there exists a number N, such that for all $n>N$, $|f_n(x)-f(x)|<\episilon$
https://en.wikipedia.org/wiki/Uniform_convergence