**Solution 1. **

**class=" fc-falcon">§ 7. **

**[Math] Why are monotone functions Riemann integrable on a closed interval [Math] Non-integrable function that has an antiderivative. The integral as the area of a region under a curve. **

**I know that bounded functions on compact intervals $[a,b]$ with only finitely (or countably) many discontinuities are Riemann integrable. **

**Let S ⊂ R be the set of discontinuities of f. **

**. Ask Question Asked 10 years, 8 months ago. 10. **

**Chapter 8 is in a sense the end of the ‘core’ part of the book, treating sequences and series of functions, power series, Taylor series and the like. **

**. Since the interval (0,1) is bounded, the function is Lebesgue integrable there too. . **

**. For each s not in S, ﬁnd a bounded continuous f for which the Lebesgue integral fails to exist. **

**But is it possible to solve this problem similarly to finite discontinuity cases?. **

**b) Find an example to show that gmay fail to be integrable if it di ers from f at a. **

**Modified 6 years, 3 months ago. We have seen that continuous functions are Riemann integrable, but we also know that certain kinds of discontinuities are allowed. **

**$\begingroup$ No, this answer is mistaken: a function may very well have infinitely many discontinuities in any subinterval of its domain of definition, and still be Riemann-integrable; the only condition is that the set of discontinuitites should have Lebesgue measure $0$. . **

**So something has to give.**

**Let g: [a;b] !R be a function which agrees with fat all points in [a;b] except for one, i. **

**. **

**Edit: Monotone function is not only increasing. However, it turns out that this function is integrable on. There are, however, many other types of integrals, the most important of which is the Lebesgue integral. **

**10. 3 uses the notion of partitions which are nite sets. For instance, a function which is $0$ on $[0,1]$, $1/(x-1)$ on $(1,2]$ is a nonnegative function, continuous except at one point, and is not even Lebesgue. . fc-falcon">extended to nitely many discontinuities. **

**$\endgroup$ –. **

**. The Lebesgue integral allows one to integrate unbounded or highly discontinuous functions whose Riemann integrals do not exist, 205. **

**For each s not in S, ﬁnd a bounded continuous f for which the Lebesgue integral fails to exist. **

**Since the interval (0,1) is bounded, the function is Lebesgue integrable there too. **

**Let f (x) be a defined function on [a, b]. **

**5 0. **

**That is S = {x ∈ R: o(f, x) > 0}. **

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