Continuity of one function, implies continuity of another?

[bobby2k]

Hi

Lets say that f(x) is continuous. Then $\int_0^x \! f(t)dt=G(x)$ is continuous. (I don't think you have to say that f need to be continuous for this, all we need to say is that f is integrable?, or do we need continuity of f here?)

But my main question is about the converse. let's say that $\int_0^x \! f(t)dt=G(x)$ is continuous, does that imply that f is continuous?

Have a nice sunday.

 

[pasmith]

Hi

Lets say that f(x) is continuous. Then $\int_0^x \! f(t)dt=G(x)$ is continuous. (I don't think you have to say that f need to be continuous for this, all we need to say is that f is integrable?, or do we need continuity of f here?)

This is true for any integrable function.

But my main question is about the converse. let's say that $\int_0^x \! f(t)dt=G(x)$ is continuous, does that imply that f is continuous?

No: by the above, $f$ does not need to be continuous for its integral to be continuous.

 

[bobby2k]

This is true for any integrable function.



No: by the above, $f$ does not need to be continuous for its integral to be continuous.

Hehe, ofcourse, thanks.