Analysis: Is f(x) Continuous at Every Number?

[AlexHall]

I need to show if the following is true or false.

If the function f: (0,1)--> R is continuous in every irrational number x then f is continuous at every number.

Thank you

 

[e(ho0n3]

The answer is false: Define f by f(x) = 0 if x is irrational; and f(x) = 1/q where x = p/q, p coprime to q. This function is continuous at every irrational and discontinuous at all the rationals.

 

[icantadd]

The answer is false: Define f by f(x) = 0 if x is irrational; and f(x) = 1/q where x = p/q, p coprime to q. This function is continuous at every irrational and discontinuous at all the rationals.

There are functions where this is true. f(x) = x is continuous at every irrational number on [0,1], and is continuous at every number. However, this is not generally the case:
I believe Thomae's function serves as a counterexample to the statement which you need to disprove which is what e(ho0on3 mentioned.

There is an interesting paper on Thomae's function by Dr. Beanland, discussing how to modify the function so that is differentiable
<www.people.vcu.edu/~kbeanland/Papers/ThomaesFunction.pdf>
Very interesting, he mentions some way to quantify irrationalness which is the part of the paper that goes over my head, but is interesting nonetheless