What is are conditions for a function to be continuous on AB?

1. What is the mathematical expression for the definition of continuity? Clarification: A function f defined on (a,b) is said to be continuous on (a,b) if it is continuous at every point of (a,b) i.e., if limx→c⁡f(x)=f(c) ∀ c ∈ (a,b).

Is a continuous function on an open interval is integrable?

Yes. More precisely: if a function is continuous then the Riemann integral exists.

Are integrable function in AB is discontinuous for?

Discontinuous functions can be integrable, although not all are. Specifically, for Riemann integration (our normal basic notion of integrals) a function must be bounded and defined everywhere on the range of integration and the set of discontinuities on that range must have Lebesgue measure zero.

How do you know if a polynomial is continuous?

Let a ∈ R be a constant, and let f be a function defined on an open interval containing a. We say f is continuous at a if limx→a f(x) = f(a). This is roughly equivalent to saying that a function is continuous if its graph can be drawn without lifting the pen.

Are rational functions continuous?

Every rational function is continuous everywhere it is defined, i.e., at every point in its domain. Its only discontinuities occur at the zeros of its denominator.

Why continuous function is integrable?

If f is continuous everywhere in the interval including its endpoints which are finite, then f will be integrable. A function is continuous at x if its values sufficiently near x are as close as you choose to one another and to its value at x.

Can I integrate a discontinuous function?

We evaluate integrals with discontinuous integrands by taking a limit; the function is continuous as x approaches the discontinuity, so FTC II will work. When the discontinuity is at an endpoint of the interval of integration [a,b], we take the limit as t approaces a or b from inside [a,b].