How to check the continuity

• Nov 12th 2011, 03:18 AM
deepak
How 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 AM
sbhatnagar
Re: How to check the continuity
Any function $y=f(x)$ is continuous at point $x=a$ if the following three conditions are satisfied :

1) $f(a)$ is defined.

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

3) $\lim_{x \to a}f(x)=f(a)$
• Nov 12th 2011, 06:15 AM
skeeter
Re: How to check the continuity
Quote:

Originally Posted by deepak
How to check the continuity of function
f(x) = x when x is rational
2-x when x is irrational

Math Tutor - Functions - Theory - Elementary Functions
• Nov 12th 2011, 11:43 AM
HallsofIvy
Re: 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 PM
Deveno
Re: 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?).