Show that if f ∈C^0(E) and E is compact, then f is uniformly continuous. (Hint: Use the fact tha

Show that if f ∈C^0(E) and E is compact, then f is uniformly...

Key Concepts

Finite Coverings

The concept of finite coverings stems from the property of compactness; any open cover of a compact set can be reduced to a finite subcover. This is essential in proofs that require uniform control over the function's behavior since it allows the domain to be partitioned into finitely many regions where the function's oscillation can be uniformly bounded.

Epsilon-Delta Definition of Continuity

The epsilon-delta definition of continuity states that at every point in the domain, given any ? > 0, there exists some ? (which may depend on the point) such that all points within ? of the original point are mapped to within ? of the function's value at that point. When combined with compactness, this local continuity condition can be elevated to a uniform one, ensuring that a single ? works for the entire domain.

Uniform Continuity

Uniform continuity is a strengthening of the usual notion of continuity, requiring that for every ? > 0 there exists a ? > 0 such that for all points x and y in the domain, if the distance between x and y is less than ?, then the distance between f(x) and f(y) is less than ?. The key aspect is that ? depends only on ? and not on the particular points selected, ensuring consistent behavior across the entire domain.

Compactness

Compactness is a topological property that, in the context of metric spaces, is equivalent to a set being both closed and bounded. Compact sets have the crucial property that every open cover has a finite subcover. This allows local properties, like continuity, to be extended to the entire set by considering finitely many neighborhoods.

Transcript

0:00 Okay.

00:02 So what we want to do is we specifically know that f is said to be uniformly continuous on a close interval from a to b.

00:22 If for some epsilon greater than zero, then there is, um, there is a close interval from a to b.

00:32 Greater than 0 such that if we have x1 and x2 on that close interval and the difference or the absolute value difference is less than than the function evaluated the difference of the two functions evaluated or the function evaluated or the function evaluated and the function evaluated and those two x values is less than epsilon.

01:12 And we want to use this.

01:16 We want to use this concept to show that the upper limit minus the lower limit is less than or equal to that epsilon times b minus a.

01:38 And so what i'm going to do is kind of do it graphically for us a little bit.

01:47 And so let's go ahead and draw a graph.

01:58 And so i'm just going to draw a graph.

02:00 And it doesn't matter what type of function you draw.

02:03 We're just going to draw some function.

02:08 And this is from a to b.

02:14 And then we are, and we're going to do it for equal with sub intervals or partitions.

02:24 It doesn't necessarily have to be that, but we're going to go ahead and do it equal with partitions.

02:30 And so i'm going to draw the upper sum, and so the upper sum rectangles.

02:46 And so there's the first one.

02:49 We're going to draw the second one.

02:52 We're trying to make them as straight as possible.

02:55 Don't.

02:56 That doesn't always work out for me, but especially on this.

03:01 And then, of course, this upper sum.

03:11 And then at some point, this is going to have to start going the other way.

03:26 And so there is, i probably should have made a more defined graph, but here.

03:32 Here is our upper sums.

03:36 And so those are our upper sums.

03:39 And now rectangles.

03:40 And now we're going to draw in our lower sum rectangles.

03:43 And so our lower sum rectangles, the top would be right there.

03:49 And, of course, the bottom would follow the length.

03:52 So all of my lower sum rectangles would be these rectangles.

04:05 And the red parts are the top of those rectangles.

04:10 And so what if i do graphically do the upper rectangles minus the lower rectangles...

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