Absolute value of a function integrable?

In summary, the statement being proven is that if a function f is continuous on the interval (a,b] and the absolute value of f is bounded on the closed interval [a,b], then f is integrable on the closed interval [a,b]. To prove this, we use the upper and lower bounds and the fact that f is continuous to create an open cover of [a,b], which is a compact set. From there, we can choose a finite subcover and use any partition that is at least as fine as the open cover to show that the upper and lower sums are less than epsilon, thus proving that f is integrable on [a,b].
  • #1
tomboi03
77
0
this is the question,
Prove that if f is continuous on (a,b] and if |f| is bounded on [a,b] then f is integrable on [a,b]. (note: it is not assumed that f is continuous at a.)

I know you have to use the upper and lower bounds to prove this statement but i don't know where to start?

Thanks
 
Last edited:
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  • #2
let eps>0
H(x):={y!=x| |f(x)-f(y)|<eps/(b-a)}
Union[H(x)|x in [a,b]]
is an open cover (by continuity of f) of [a,b] a compact set so we may chose a finite subcover
is P is any partition at least as fine as the open cover will have
U-L<eps
qed
 

1. What is the definition of the absolute value of a function?

The absolute value of a function is a measure of its distance from the origin on a graph. It is always a positive value and is represented as "|f(x)|".

2. How do you determine if a function is integrable?

A function is integrable if it has a finite area under the curve and does not have any discontinuities or infinite values within the interval of integration.

3. Can a function with a discontinuity be integrable?

No, a function with a discontinuity is not integrable as it does not have a finite area under the curve.

4. How does the absolute value of a function affect its integrability?

The absolute value of a function can affect its integrability by changing the shape of the graph and potentially creating discontinuities. However, as long as the function remains continuous and has a finite area under the curve, it can still be integrable.

5. Are there any special techniques for integrating absolute value functions?

Yes, there are specific techniques for integrating absolute value functions, such as splitting the integral into separate parts based on the intervals where the function is positive and negative. Additionally, using symmetry and substitution can also be helpful in integrating absolute value functions.

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