Discuss the continuity of the function defined by:

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- Feb 2nd 2010, 09:43 PMvuze88Continuity Question
Discuss the continuity of the function defined by:

- Feb 2nd 2010, 10:08 PMProve It
- Feb 2nd 2010, 10:16 PMvuze88
Can you explain why every rational number is between two irrational numbers? Also, according to the textbook, it is discontinuous everywhere but is continuous at 0.

- Feb 3rd 2010, 05:58 AMHallsofIvy
Yes, the function is continuous at 0. Given any > 0, [tex]|f(x)- f(0)| is |x- 0|= |x| if x is rational, and |-x- 0|= |x| if x is irrational. Take to be any number less than .

If a is any rational number other than 0, f(x)= a so |f(x)- f(a)= |x-a| if x is rational and |-x-a|= |x+a| if x is irrational. No matter how small is, there will exist both rational and irrational numbers between and so with less than half the difference between them, there will exist x such that |f(x)- f(a)|> [tex]\epsilon[/b]. The same is true if a is irrational.