Continuity of Dirichlet looking function

In summary, the author is trying to solve a problem where the function is not continuous at any rational number less than or equal to c. If he solves the problem for all rational numbers less than or equal to c, then the function is continuous at c. However, if he solves the problem for all irrational numbers less than or equal to c, then the function is not continuous at c.
  • #1
teleport
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Homework Statement



Where is the function f(x) continuous?

f(x) =
x, if x is rational

0, if x is irrational

Homework Equations





The Attempt at a Solution



Is this correct?: I approach some c =/= 0, 1st through x's that are rational
and prove there is the limit c, and then approach through x's that are irrational and prove that the limit now cannot be c, that now I can conclude that the limit at c does not exist, and hence the function is not continuous at any c=/=0?

If no, why, and in what other way must I solve it then? If yes, please try to explain as rigourously as you can why this can be done. Remind you though, don't get technical above 2nd year in which I am. Thank you.
 
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  • #2
You have it right. If you approach through irrationals, the limit is always zero. Which is not equal to c unless c=0. There's not really much to get technical about beyond the statement that both the rationals and irrationals are dense in R.
 
  • #3
Well you just did! Sorry, but by dense you mean that: for any pair of irrationals there is a rational between them, and the same the other way? This is what I vaguely remember I read some day on the web. How is it that dense implies you can do that? Thanks again.
 
  • #4
A set S is dense if for any x you can find an s in S that is as close to x as you want. Which would let you say for any x you can find a sequence of rationals approaching it and a sequence of irrationals. Do you think you are expected to prove that?
 
  • #5
No way! But how is it that the rationals and irrational are proven to be dense?
 
  • #6
just basically showing that the limit as s in S approaches x is x itself, and that's it?
 
  • #7
The proof depends on how you define an irrational number and can be sort of 'technical'. But try this, if x is rational, then {x+sqrt(2)/n} is a sequence of irrationals approaching x. If you think of an irrational in terms of it's decimal expansion then for example sqrt(2) is the limit of the sequence 1,1.4,1.41,1.414,... Each term in the series adds one more decimal place to the expansion and each term in the series is rational. How's that for 2nd year level?
 
  • #8
Sweet! :) Even my neighbor can get that. I'll go there now and show it to him (joking). Thanks so much.
 

FAQ: Continuity of Dirichlet looking function

What is the "Continuity of Dirichlet looking function"?

The Continuity of Dirichlet looking function is a mathematical concept that determines the smoothness and connectedness of a function at different points. It is a measure of how the values of a function change when the input values change.

Why is the "Continuity of Dirichlet looking function" important in mathematics?

The Continuity of Dirichlet looking function is important because it helps us understand the behavior of a function and its limits. It also allows us to make predictions and solve problems in various fields of mathematics, such as calculus, analysis, and differential equations.

How is the "Continuity of Dirichlet looking function" calculated?

The Continuity of Dirichlet looking function is calculated using the concept of limits. A function is considered continuous if the limit of the function at a point exists and is equal to the value of the function at that point. This means that as the input values approach a certain point, the output values also approach a certain value.

What is the difference between continuous and discontinuous functions?

A continuous function has no abrupt changes or breaks and can be drawn without lifting the pen from the paper. On the other hand, a discontinuous function has breaks, jumps, or holes in its graph, which means that the limit of the function at a point does not exist.

How can the "Continuity of Dirichlet looking function" be applied in the real world?

The Continuity of Dirichlet looking function can be applied in various real-world scenarios, such as predicting the growth of populations or the spread of diseases, analyzing the flow of water in rivers, and understanding the behavior of stock prices in the stock market. It is also crucial in engineering and physics, where continuous functions are used to model real-world phenomena.

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