I have a problem and a solution here please let me know

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The discussion centers on the integrability of the function 1/f, where f is defined on the interval [a,b] and is integrable and bounded below by a positive constant t. The claim is that 1/f is also integrable, supported by the argument that since f is bounded above and below, 1/f must also be bounded. However, a counterpoint is raised regarding the continuity of 1/f, emphasizing the need to consider the behavior of f in different segments of the interval, particularly when f approaches zero. The conversation highlights the importance of understanding the properties of functions in relation to their integrability.

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I have a problem here and a solution. Not sure if I am on the right track. Please let me know if this proof is right.

we are given:f :[a,b]-R, f is integrable and bdd below(f is greater
than t(i use t instead of delta) for all x belongs to [a,b] ), t is
greater than 0
claim: 1/f is integrable.
solution: f is integrable implies f is bdd - f is bdd above and below
implies 1/f is bdd below and above - 1/f is bdd-{1}
f is greater than t and t is greater than 0 for all x
- f is greater than 0 for all x. - 1/f can not be infinity for all x-
1/f is continuous.{2}
from {1} & {2} 1/f is a bdd continuous function .
hence 1/f is integrable.
 
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How can you conclude that 1/f is continuous when you don't even know that f is?
 
f might not be bounded. I would consider the integral in two parts: One where f<2t and the other where f>= 2t (say). That is one integral where f is big and one where f is not so big.

As far as how to split up the interval that way would depend on the kind of integral you are doing. Lebesgue - no problem: If you are only dealing with Riemann - may have to turn to (piecewise) continuity to separate the interval accordingly.
 
f(x) identically equal to 0 satisfies all the conditions you give but 1/f certainly is not bounded!
 
Halls, one of the conditions was f(x)>t>0 for all x in [a,b].
 
what is bdd?
 
murshid islam: bdd= bounded.

matt grime- I really do need to learn to read, don't I!
 

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