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

Since the interval (0,1) is bounded, the **function** is Lebesgue **integrable** there too. 6.

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

7) Assume f: [a;b] ! R is **integrable**.

Since there are plenty of sets that are uncountable with measure zero (like the Cantor set), the answer to your questions is yes a** Riemann integrable function** can have uncountably many** discontinuities** in its domain of. . Then the set of points, at which it is discontinuous, is countable.

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

Then f is Riemann **integrable** if and only if the set of **discontinuities** of f is of measure zero (a null set). If you look for. 5 0.

. .

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

class=" fc-falcon">6.

. **Functions with** finitely **many** removable **discontinuities** are **integrable**.

. Modified 6 years, 3 months ago.

**integrable**

**function**.

An example of** discontinuous integrable function. **

**. **

**). 1. Hence, Dirichlet’s Function is continuous on [0;1] except on a set of De nitioncontent zero. **

**. We abbreviate the phrase almost everywhere by writing a. 3: Integrating Functions with Discontinuities Example The function f(x) := (cos(1/x) if x̸= 0 , 0 if x= 0, is discontinuous at x= 0 but continuous at all x̸= 0. I think they are removable, because the discontinuities can be removed by redefining the. . class=" fc-falcon">Riemann integral. **

**In Chapter 5 we defined the Riemann integral of a real function f over a bounded. **

**. . **

**7) Assume f: [a;b] ! R is integrable. **

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**5 0. **

**there are uncountably many functions which are not analytically representable in Baire’s. **

**This is the first video in the Lebesgue's Criterion for Riemann Integrability series. **