Muddiest Point 10/19/17
I think the muddiest point from Thursday’s class is the distinction between Uniform continuity and point-wise continuity.
First, we recall the definitions of pointwise continuity and uniform continuity.
Uniform continuity: A function f : A → R is uniformly continuous on A if for every ε > 0 there exists a δ > 0 such that for all x, y ∈ A, |x − y| < δ implies |f(x) − f(y)| < ε.
Point-wise continuity: A function f : A → R is continuous at a point c∈A if,for all ε>0,there exists a δ>0 such that whenever |x−c|<δ (and x ∈ A) it follows that |f(x) − f(c)| < ε.
With uniform continuity, the choice for δ depends only on ε whereas when showing a function is continuous at a point c. the choice for δ is allowed to depend on both ε and c.
To better visualize the difference, I would like to demonstrate them in images.
Point-wise continuity: A function is continuous if, for a given point p0, for any ε, we can find some δ, such that the distance between the points (p0 and p in the picture)must be less than δ and the distance of images is less than ε.
Uniform continuity: We need to guarantee the two points in the δ range, the images of the two points will be fall in the ε range. For uniform continuity, we have to show this for every two points in the function.
In addition, uniform continuity is a strictly stronger property. If f is “uniformly continuous on A”, ε and δ can be chosen that works simultaneously for all points c in A.
If a function f is not uniformly continuous on a set A, it tells us that there is some ε > 0 for which no single δ > 0 satisfies that the images of the two points fall into the ε range for all c ∈ A.
Good elaboration on this muddy point. Thank you, Shuyi.