2 questions on continuity/continuous extensions

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In summary, if D is a subset of R and f is uniformly continuous, then f has a continuous extension to R.
  • #1
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If D is a dense subset of R and f is uniformly continuous, prove that f has a continuous extension to R.

I said:

if x0 is in D, then f(x0) is continuous.

Let x0 be in R\D. D dense in R--> there exists {x_n} in D s. t. {x_n}--> x0.

{x_n} is a Cauchy sequence converging to x. f is uniformly continuous on D--> {f(x_n)} is Cauchy and thus converges to y.

Define f(x0)=y

Then I'm stuck. I want to show that f(x0) is continuous, but I'm not sure how.

Another question: Prove that if D is closed (instead of dense) , then f also has a continuous extension

I'm not sure even where to begin with this question.
 
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  • #2
For the first one, just use the definition of continuity. For the second the extension is clearly not unique so you just need to define one. Hint: the complement of a closed set in R is a countable union of open intervals.
 
  • #3
akoska said:
If D is a dense subset of R and f is uniformly continuous, prove that f has a continuous extension to R.

I said:

if x0 is in D, then f(x0) is continuous.
f(x0) is a number. It makes no sense to say that a number is continuous. Perhaps you meant to say that f is continuous at x0.

Let x0 be in R\D. D dense in R--> there exists {x_n} in D s. t. {x_n}--> x0.

{x_n} is a Cauchy sequence converging to x. f is uniformly continuous on D--> {f(x_n)} is Cauchy and thus converges to y.
Do you need uniformly continuous for that?

Define f(x0)=y

Then I'm stuck. I want to show that f(x0) is continuous, but I'm not sure how.

Another question: Prove that if D is closed (instead of dense) , then f also has a continuous extension

I'm not sure even where to begin with this question.
 
  • #4
HallsofIvy said:
f(x0) is a number. It makes no sense to say that a number is continuous. Perhaps you meant to say that f is continuous at x0.


Do you need uniformly continuous for that?


yes, sorry, that's what I meant.

No, you don't. I simply did not want to type an almost identical question out again, and then forgot to mention that you don't need uniform continuity.

So, using the definition of continuity, every sequence {x_n} in D that converges, {f(x_n)} converges. But this only takes into account {x_n} in D, right? So what about {x_n} in R? Is it that: Given a converging sequence {x_n} in R that converges to x0, we can find an equivalent sequence {x_n} in D that converges to x0 because x0 must be a limit point of D or a point in D (consequence of D dense)? So f is continuous on R.

I'm still unclear on what to do in the closed set quesion.
 
  • #5
Because D is dense if you have a cauchy sequence if R, you can find a cauchy sequence in D with the same limit. In light of what I said about the second question you only have to define f in a continuous way across open intervals. If f(0)=0 and f(1)=1 how would you define f on (0,1) to make it continuous?
 

What is continuity?

Continuity is a mathematical concept that describes a function's ability to have a connected graph without any breaks or abrupt changes. A continuous function can be drawn without lifting the pen from the paper.

What is a continuous extension?

A continuous extension is a function that extends the domain of a given function while maintaining its continuity. It is used to bridge any gaps or breaks in the original function's domain, allowing it to be continuous over a larger interval.

How do you determine if a function is continuous?

A function is considered continuous if it satisfies three conditions: 1) the function is defined at the point in question, 2) the limit of the function at that point exists, and 3) the limit and the function's value at that point are equal.

Can a function have a continuous extension at a point where it is not originally defined?

Yes, a function can have a continuous extension at a point where it is not originally defined. This is done by defining the function at that point using a limit, or by using a piecewise function that is continuous at that point.

Why is continuity important in mathematics?

Continuity is important in mathematics because it allows us to make predictions and analyze functions with ease. It also helps us understand the behavior of a function and its properties, making it a fundamental concept in calculus and other areas of mathematics.

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