Thread: 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

Any function is continuous at point if the following three conditions are satisfied :

1) is defined.

2) exists i.e is finite.

3)

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

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?

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?).