## Is every integrable function is continuous?

Continuity implies integrability; if some function f(x) is continuous on some interval [a,b], then the definite integral from a to b exists. While all continuous functions are integrable, not all integrable functions are continuous.

**Can you integrate non continuous functions?**

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].

### Can a non continuous function be Riemann integrable?

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.

**Can you take the Antiderivative of a non continuous function?**

No. The best you can do is a function g that is continuous everywhere, and differentiable with g′(x)=f(x) for all x except at x=0. Since g′(x) has different one sided limits at x=0, g cannot be differentiable there.

## Is every continuous function Riemann integrable?

Theorem. All real-valued continuous functions on the closed and bounded interval [a, b] are Riemann- integrable.

**Is every differentiable function integrable?**

Well, If you are thinking Riemann integrable, Then every differentiable function is continuous and then integrable! However any bounded function with discontinuity in a single point is integrable but of course it is not differentiable!

### Are there functions that don’t have integrals?

Definitely not. Simple examples include things like e-x2 and xx. Of course, these functions do have integrals – they’re smooth and continuous and all that, so they have areas under their curves – so (using the fundamental theorem of calculus) you can use those areas to construct, at each point, its antiderivative.

**Which is an example of a non-continuous, integrable function?**

You can use these theorems to give examples of noncontinuous integrable functions. By 1 and 3, any function that’s continuous except at finitely many places is integrable. For example, the signum function is not continuous at 0… Loading…

## How is integrability related to the property of continuous?

It’s enough to change the value of a continuous function at just one point and it is no longer continuous. Integrability on the other hand is a very robust property. If you make finitely many changes to a function that was integrable, then the new function is still integrable and has the same integral.

**Can a definite integral be a non integrable function?**

The integral can be defined as an improper Riemann integral and happens to equal π. This is just a warning that ‘not integrable’ (in the Lebesgue sense) doesn’t mean that the definite integral doesn’t have a value. ( x) / x and thus computing the integral over an interval using the fundamental theorem of calculus.

### Can a continuous function be a Riemann integrable?

every continuous function is Riemann integrable,continuity is certainly not necessary. I dont know anything about measure. Define f to be identically 0 on R, except that f ( 0) = 1. To see that this is Riemann integrable, note that the lower sums are all 0 (suppose we’re integrating on [ − 1, 1], for clarity).