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Taylor King
Answered 2020-12-15 22:41:24

um I'm confused

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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.


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.


An antiderivative, F, is normally defined as the indefinite integral of a function f. F is differentiable and its derivative is f.If you do not assume that f is continuous or even integrable, then your definition of antiderivative is required.


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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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No, all functions are not Riemann integrable


No, because there is no greatest integer.


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).


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


A linear function is any function that graphs to a straight line. What this means mathematically is that the function has either one or two variables with no exponents or powers. If the function has more variables, the variables must be constants or known variables for the function to remain a linear function.


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