Was wondering if someone could outline the following problem:

Let g: R to R by defining g(x) as 2x for x rational and as x+3 for x irrational. Find all the points where g is continuous and prove your assertion.

Thanks for any help.

Jack

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- November 11th 2006, 11:07 AMJackStolermanAnalysis Countinuity Problem
Was wondering if someone could outline the following problem:

Let g: R to R by defining g(x) as 2x for x rational and as x+3 for x irrational. Find all the points where g is continuous and prove your assertion.

Thanks for any help.

Jack - November 11th 2006, 11:54 AMPlato
Do you know how to solve 2x=x+3?

- November 11th 2006, 01:15 PMJackStolerman
Well how would I prove the assertion?

- November 11th 2006, 01:34 PMPlato
Well prove that it is continuous only at 3.

- November 11th 2006, 01:47 PMThePerfectHacker
Basically the Dirchelet function.

When we approach a number (limit at the point) by rationals we get one value and when we approach a number (limit at the point) by irrationals we might get a different value. Well, approaching a number (limit) by rationals is anagolous with the irrationals because the irrationals are contrsucted by a Cauchy sequence of rationals. Thus, what Plato said is to solve the equation, for it says the rationals and irrationals produce the same limit (which must be if it exists).