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Not every function with discontinuities is integrable. A function is considered integrable (in the Riemann sense) if the set of its discontinuities has measure zero. Functions with too many or too "wild" discontinuities, such as the Dirichlet function (which is 1 for rational numbers and 0 for Irrational Numbers), are not Riemann integrable. However, they may still be Lebesgue integrable under certain conditions.
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Is the greatest integer function x integrable over the real line?
yes
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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What are the discontinuities of the function f(x) the quantity x squared minus 16 over the quantity 4x minus 24?
To find the discontinuities of the function ( f(x) = \frac{x^2 - 16}{4x - 24} ), we first identify the points where the denominator is zero. Setting ( 4x - 24 = 0 ) gives ( x = 6 ). The numerator ( x^2 - 16 ) can be factored as ( (x - 4)(x + 4) ), but it does not affect the discontinuity since it does not equal zero at ( x = 6 ). Therefore, the function has a discontinuity at ( x = 6 ).
Is x2 an acceptable wave function?
The function (x^2) is not an acceptable wave function because it does not satisfy the normalization condition required for quantum mechanical wave functions. A valid wave function must be square-integrable over its domain, meaning that the integral of the absolute square of the function over all space must be finite. In the case of (x^2), the integral diverges, making it non-normalizable and therefore not a valid wave function.
Is the relation a function?
Not every relation is a function. But every function is a relation. Function is just a part of relation.
What function is integrable but not continuous?
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 continuous function is integrible but converse is not true every integrable function is not continuous?
That's true. If a function is continuous, it's (Riemman) integrable, but the converse is not true.
Are continuous functions integrable?
yes, every continuous function is integrable.
Can you Give an example of bounded function which is not Riemann integrable?
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.
What are discontinuities?
Discontinuities in mathematics refer to points on a function where there is a break in the graph. They can occur when the function is not defined at a particular point or when the function approaches different values from the left and right sides of the point. Common types of discontinuities include jump discontinuities, infinite discontinuities, and removable discontinuities.
Is the greatest integer function x integrable over the real line?
yes
Discontinuities of first and second kind?
A discontinuity of the first kind occurs when a function's limit does not exist at a specific point, while a discontinuity of the second kind happens when the function's value at a particular point is undefined or infinite. Discontinuities of the first kind can be classified as removable, jump, or infinite discontinuities, based on the behavior of the limit.
Let f be an odd function with antiderivative F. Prove that F is an even function. Note we do not assume that f is continuous or even 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.
Is every measurable functions continuous?
No, a measurable function may have a finite number of discontinuities (for the Riemann measure), or a countably infinite number of discontinuities (for the Lebesgue measure). It should also be bounded (have some upper and lower bound, or limit, in the domain that is being measured), to be measureable. At least, some unbounded functions are not measurable.No, a measurable function may have a finite number of discontinuities (for the Riemann measure), or a countably infinite number of discontinuities (for the Lebesgue measure). It should also be bounded (have some upper and lower bound, or limit, in the domain that is being measured), to be measureable. At least, some unbounded functions are not measurable.No, a measurable function may have a finite number of discontinuities (for the Riemann measure), or a countably infinite number of discontinuities (for the Lebesgue measure). It should also be bounded (have some upper and lower bound, or limit, in the domain that is being measured), to be measureable. At least, some unbounded functions are not measurable.No, a measurable function may have a finite number of discontinuities (for the Riemann measure), or a countably infinite number of discontinuities (for the Lebesgue measure). It should also be bounded (have some upper and lower bound, or limit, in the domain that is being measured), to be measureable. At least, some unbounded functions are not measurable.
Can a discontinuous function be developed in a Fourier series?
Yes, a Fourier series can be used to approximate a function with some discontinuities. This can be proved easily.
How do you determine if the graph is a function?
If for every point on the horizontal axis, the graph has one and only one point corresponding to the vertical axis; then it represents a function. Functions can not have discontinuities along the horizontal axis. Functions must return unambiguous deterministic results.
What are the discontinuities of the function f(x) the quantity of x squared plus 5 x plus 6 all over 2 x plus 16.?
This is a rational function; such functions have discontinuities when their DENOMINATOR (the bottom part) is equal to zero. Therefore, to find the discontinuities, simply solve the equation:Denominator = 0 Or specifically in this case: 2x + 16 = 0
