Analysis- upper and lower integrals

[kristinv88]

Suppose that the bounded function f:[a,b]-->R has the property that for each rational number x in the interval [a,b], f(x)=o for all x in [a,b]. Prove that
the lower integral of f from a to b is less than or equal to zero which is less than or equal to the upper integral of f from a to b.

Here's what I have so far (idk if it's right though!):

Define the upper integral=sup{L(f,P) s.t. P is a partition of the interval [a,b]}.
Define the lower integral=inf{U(f,P) s.t. P is a partition of the interval [a,b]}.

Case 1: x is rational
upper integral=sup{0}=0
lower integral=inf{0}=0
so our answer for this case is trivial

Case 2: x is irrational
this is where I'm stuck!

 

[grey_earl]

Your integrals are Riemann integrals, right? (Should be from your definition). So you must consider rational and irrational values of x _at once_, there is no point in seeing them separately. Therefore, for each closed interval I in your partition P, can the minimum of { f(x): x from I } be larger than zero? Can the maximum be smaller than zero? That's all you need.

 

[kristinv88]

They are Riemann integrals. I see what you mean about not splitting it up into two cases. But I don't see why the minimum can't be larger than zero or why the maximum couldn't be less than zero

 

[grey_earl]

Each (closed) interval I contains rational and irrational numbers, right? So for each I, there are two possibilities: the minimum is at a rational number, or at an irrational one. If it is at a rational number, it is zero. If it is at an irrational one, will it be positive or negative?

 

[kristinv88]

i understand it now! thank you soo much for your help