what functions are integrable?
if f(x) is continuous on [a,b], then f(x) is integrable of [a,b]
please explain completely and other situations
What exactly do you mean? What kind of answer are you looking for?
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.
I think a function is integrable if it be dic-continus on countable point
when a function is not continuos how can it be integrable?
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.
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