Which Functions Are Integrable?

[mechanical eng]

what functions are integrable?

 

[Kuno]

if f(x) is continuous on [a,b], then f(x) is integrable of [a,b]

 

[mechanical eng]

please explain completely and other situations

 

[d_leet]

please explain completely and other situations

What exactly do you mean? What kind of answer are you looking for?

 

[Crosson]

Essentially a function is integrable if it is bounded, and discontinuous on at most a set of measure zero.

I suspect you are really asking: when can the anti-derivative of a function be expressed in terms of sums, products, powers, exponentials, trig functions, and the inverses of such. The conditions are not hard to state, but they are not interesting either.

 

[reza]

please explain completely and other situations

I think a function is integrable if it be dic-continus on countable point

 

[mechanical eng]

integral

when a function is not continuos how can it be integrable?

 

[yenchin]

Not really treating this rigorously, but the idea is simple. Say f(x) = x at all x except when x=1. Say f(1) = 5.

Then f is discontinuous at the point x=1. But if you interpret the integral as area under the graph, then it is intuitively clear that the integral of f from say, x=0 to x=2, is the same as that of the integral of the identity function from x=0 to x=2. The point at x=1 does not contribute to the area.

So as Crosson said, essentially, a function is integrable if it is bounded, and discontinuous on at most a set of measure zero.