What are the values of the integral and derivative for the Dirichlet function?

[Karlisbad]

let be the function f(x) so

f(x)=1 for every integer or rational.

f(x)=0 otherwise..

my questions are, what's the value of \int_{a}^{b}f(x)dx and f'(x) (i think the second value is 0 for every x, but I'm not sure)

 

[matt grime]

The function is neither Riemann integrable, nor differentiable. It is not even continuous os has no chance of being differentiable. It is Lebesgue integrable and the integral is identically zero.

 

[Karlisbad]

Oh..sorry, i saw a similar example, they defined the function:

g(x)= f(x)+a(x) iff x is an integer

g(x)=f(x) otherwise.

where a(x) is the function (i don't know its name) that is 1 iff x is a rational and 0 otherwise. :shy: :shy: then you had the question of defining g'(x)

 

[HallsofIvy]

Oh..sorry, i saw a similar example, they defined the function:

g(x)= f(x)+a(x) iff x is an integer

g(x)=f(x) otherwise.

where a(x) is the function (i don't know its name) that is 1 iff x is a rational and 0 otherwise. :shy: :shy: then you had the question of defining g'(x)

Obviously, g(x)= f(x) for all non-integer x so g'(x)= f'(x) for non-integer x. Since g is not continuous at integer x, g is not differentiable there.

 

[arildno]

As a completely unnecessary addition to the previous posts, the function you started with is famous enough to have gained its own name: it is called the Dirichlet function.