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Q: Let f be a bounded measurable function assume that there exist constants co and 01 such that 0mxRfxc show that f is integrable over R?

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yes, every continuous function is integrable.

That's true. If a function is continuous, it's (Riemman) integrable, but the converse is not true.

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yes.since this functin is simple .and evry simple function is measurable if and ond only if its domain (in this question one set) is measurable.

Yes. A well-known example is the function defined as: f(x) = * 1, if x is rational * 0, if x is irrational Since this function has infinitely many discontinuities in any interval (it is discontinuous in any point), it doesn't fulfill the conditions for a Riemann-integrable function. Please note that this function IS Lebesgue-integrable. Its Lebesgue-integral over the interval [0, 1], or in fact over any finite interval, is zero.

A function may have a finite number of discontinuities and still be integrable according to Riemann (i.e., the Riemann integral exists); it may even have a countable infinite number of discontinuities and still be integrable according to Lebesgue. Any function with a finite amount of discontinuities (that satisfies other requirements, such as being bounded) can serve as an example; an example of a specific function would be the function defined as: f(x) = 1, for x < 10 f(x) = 2, otherwise

Every monotonic function f is R-integrable.

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A function of the form f(x) = mx + c where m and c are constants is linear.

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If it can be written in the form y = mx + c where m and c are constants [or, equivalently, ax + by = k where a, b and k are constants] then y is a linear function of x.

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Question: What are values that are used with a function in Excel? Answer: Arguments There are several possibilities. They can be called arguments and there are two kinds, variables and constants. Variables can have different values and constants are always the same.

It can be written in the form y = ax2 + bx + c where a, b and c are constants and a â‰ 0

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No, because there is no greatest integer.

No, all functions are not Riemann integrable

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A linear function is one of the form f(x) = a*x + b where a and b are constants.

ax2 + bx + c = 0 where a, b and c are constants and a is not 0.

Yes. For every measurable function, f there's a sequence of simple functions Fn that converge to f m-a.e (wich means for each e>0, there's X' such that Fn|x' -->f|x' and m(X\X')<e).