Afunction defined on an interval [a, b] or (a, b) is uniformly continuous if for each ϵ> 0 there exists a δ> 0 such that |x−t|< δ implies that |f(x)−f(t)|< ϵ. then it gives a little note saying that δ cannot depend on x, it can only depend on ϵ.with ordinary continuity, the δ can depend on both x and ϵ. i'm just a little lost on why |x−t|< δ implies |f(x)−f(t)|< ϵ, and how δ can't depend on x but only ϵ.

Step-by-step explanation:

The secret for understanding the uniform continuity is to understand the continuity in a point. Recall, we say that a function f is continuous at a point x_0 if and only if or each ϵ>0 there exists a δ>0 such that |x_0−t|<δ implies that |f(x_0)−f(t)|<ϵ.

Consider this point of view. We are studying the continuity of a function f at x_0. The definition means that if we fix an ϵ>0, no matter how small, we can find δ>0 such that all the points t at distance δ from x_0 have the its images f(t) at a distance at most ϵ from f(x_0). In ‘‘natural’’ language. If we separate a little from the point x_0, the images of the function f do not separate too much from f(x_0).

Now, notice that we are asking this condition for a fixed value x_0. What happens if check for the continuity of f at another point x_1? Given the same ϵ, will we find the same value for δ, or will be smaller?

The idea with uniform continuity is that the same δ works for the same ϵ, independently of the point x. While continuity is a local property (at a point), uniform continuity is a global property (in all a set or interval).

Check the attached images. The function is f(x)=1/x. Notice that the red band is for all the points (in the image) at a distance less than ϵ from f(x_i), i=0,1 and the black band is for the points at distance less than δ from x_i, i=0,1. In both images the red band has the same length (more or less).

Notice that the black band around x_0 is bigger than the black band around x_1. So, when we get closer to zero, the values of δ needed to keep the values of f(t) closer to f(x) become smaller. Hence, δ depends of ϵ and x.

the data of the detroit pistons in numerical order from least to greatest is 0.30, 0.37, 0.55, 0.62, 0.69, 0.72, 0.73, 0.75, 0.77, 0.78, 0.80, 0.81, 0.81, 0.83, 0.84, 0.89

the data of the miami heat in numerical order from least to greatest is 0.50, 0.50, 0.56, 0.57, 0.67, 0.72, 0.72, 0.72, 0.75, 0.75, 0.76, 0.81, 0.82, 0.90, 1.00, 1.00

the detroit pistons median : 1.52

the miami heat median: 1.47

the iqr of this data is 0.05

my guess would be b and d, but i didn’t actually get out the pen and paper. lol