Math and Arithmetic

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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.

constants, MAX_(function), etc.

"The" exponential function is ex. A more general exponential function is any function of the form AeBx, for any non-xero constants "A" and "B". Alternately, Any function of the form CDx (for constants "C" and "D") would also be considered an exponential function. You can change from one form to the other.

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

Characteristic function of any borel set is an example of simple Borel function

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.

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.

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.

The constants effect the shifts being vertical. EX. y=x+1 Normally the function would be y=x, but the (+1) Shifts the function up 1

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

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

Cos it contains useful function-declarations, constants, types.

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.

f(x) = mx +b. m and b are arbitrary constants.

f(x) = ax + b is a linear function of x, where a and b are constants.

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