Proving Integrability of Constant Function f on [a,b]

In summary, when proving that cf is integrable on [a,b] and that \int cf = c \int f, it is helpful to treat the cases c\leq0 and c\geq0 separately. This is because the definition of \int_a^b f(x) dx may differ depending on the sign of c.
  • #1
tomboi03
77
0
Suppose that f is integrable on [a,b]. Let c E [tex]\Re[/tex]. Prove that cf is integrable on [a,b] and that
[tex]\int[/tex] cf= c [tex]\int[/tex] f
(both integrals goes from a to b i didn't kno how to put it in the integral)

I'm not sure how to go about this problem
Please help me out~
Thanks
 
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  • #2
When you are completely "stuck", look back to the definitions.

How have you defined [itex]\int_a^b f(x) dx[/itex]?
 
  • #3
okay, i just got a hint saying that it would be best if we treat this separately with the cases c[tex]\leq[/tex]0 and c[tex]\geq[/tex]0.

how would that help me out? wouldn't it just be the same thing except it will be either negative or positive?
 
  • #4
How have you defined [itex]\int_a^b f(x)dx[/itex]?
 

1. What does it mean for a function to be integrable?

Integrability refers to the ability to find the definite integral of a function between two given limits. It is a measure of how well-behaved a function is and whether it can be represented by a finite area under its curve.

2. How do you prove that a constant function is integrable on a closed interval?

To prove that a constant function f on [a,b] is integrable, we need to show that it satisfies the necessary and sufficient condition for integrability, which is the Riemann Integrability Criterion. This criterion states that a function is integrable if and only if it is bounded and the set of discontinuities has a measure of zero. Since a constant function is always bounded and has no discontinuities, it is integrable on any closed interval [a,b].

3. What is the significance of proving integrability of a constant function?

Proving the integrability of a constant function may seem trivial, but it is an important step in understanding the fundamentals of integration. It helps establish the basic properties of integrals and lays the foundation for more complex integration techniques.

4. Can a constant function be integrable on an open interval?

No, a constant function cannot be integrable on an open interval as it does not satisfy the Riemann Integrability Criterion. Since an open interval has infinite endpoints, the function cannot be bounded, and thus, cannot be integrable.

5. Are there any other methods for proving the integrability of a constant function?

Yes, there are other methods for proving integrability, such as the Lebesgue Integrability Criterion or the Cauchy Criterion. However, these methods are more advanced and are not necessary for proving the integrability of a simple constant function on a closed interval.

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