How to check the continuity of function

f(x) = x when x is rational

2-x when x is irrational

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- Nov 12th 2011, 03:18 AMdeepakHow to check the continuity
How to check the continuity of function

f(x) = x when x is rational

2-x when x is irrational - Nov 12th 2011, 05:59 AMsbhatnagarRe: How to check the continuity
Any function $\displaystyle y=f(x)$ is continuous at point $\displaystyle x=a$ if the following three conditions are satisfied :

1)$\displaystyle f(a)$ is defined.

2)$\displaystyle \lim_{x \to a}f(x)$ exists i.e is finite.

3)$\displaystyle \lim_{x \to a}f(x)=f(a)$ - Nov 12th 2011, 06:15 AMskeeterRe: How to check the continuity
- Nov 12th 2011, 11:43 AMHallsofIvyRe: How to check the continuity
And, of course, there exist both rational and irrational numbers arbitrarily close to any number. So you can take the limit at any number by a sequence of rational numbers or a sequence of irrational numbers. I think you will find that this function is continuous at exactly one value of x. What is that value of x?

- Nov 12th 2011, 02:03 PMDevenoRe: How to check the continuity
here is a hint: at any real number a, let ε = |1-a|. use the definition of continuity.

(there is one point, at which you'll have to choose some other ε. why is this not a problem?).