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## Main Question or Discussion Point

While reading my calc book, I have developed a few questions about the situations in which definite integrals can exist. I've thought about these questions, and I feel that if I am able to answer some of them, I can make some other problems much easier, such as testing for convergence of a definite integral.

1) If a definite integral diverges, can we conclude that it does not exist?

2) If a definite integral does not exist, can we conclude that it diverges?

3) If [itex] f(a) [/itex] is not defined, does [itex]\int^{b}_{a}f(x)dx [/itex] necessarily not exist?

3) If [itex] f(a) [/itex] is not defined, but but [itex]\lim_{x→a+}f(x) [/itex] is, does [itex]\int^{b}_{a}f(x)dx [/itex] necessarily not exist?

5) If [itex] f(a) [/itex] is defined, but [itex]\lim_{x→a+}f(x) [/itex] does not, does [itex]\int^{b}_{a}f(x)dx [/itex] necessarily not exist?

I assume all these questions can be answered using the definition of the definite integral, along with the definition of continuity, but how exactly?

Thanks!

Note: Assume that b>a for the above integrals

BiP

1) If a definite integral diverges, can we conclude that it does not exist?

2) If a definite integral does not exist, can we conclude that it diverges?

3) If [itex] f(a) [/itex] is not defined, does [itex]\int^{b}_{a}f(x)dx [/itex] necessarily not exist?

3) If [itex] f(a) [/itex] is not defined, but but [itex]\lim_{x→a+}f(x) [/itex] is, does [itex]\int^{b}_{a}f(x)dx [/itex] necessarily not exist?

5) If [itex] f(a) [/itex] is defined, but [itex]\lim_{x→a+}f(x) [/itex] does not, does [itex]\int^{b}_{a}f(x)dx [/itex] necessarily not exist?

I assume all these questions can be answered using the definition of the definite integral, along with the definition of continuity, but how exactly?

Thanks!

Note: Assume that b>a for the above integrals

BiP