For what values of x the given function is continuous f/x 0 if x is rational 1 if x is irrational?

If you changed the function so that h(x) = 0 when x is irrational and h(x) = x when x is rational, then h would be continuous at 0, since for values b that are near 0, h(b) would be near h(0). Unfortunately, 0 is the only value of x at which h is continuous.

Is the function f/x )= 0 if x is irrational f/x )= 1 Q If X is rational x p q )] continous in R and Q?

If x = p/q, then 1/q = x/p. So f(x) = x/p if x is rational. For x=0, p = 0. This means that f(0) = 0/0.

Can X be irrational in a function?

1. Every real number x is either rational or irrational (but it can’t be both.)

For what values of x is f/x continuous?

A function continuous at a value of x. is equal to the value of f(x) at x = c. then f(x) is continuous at x = c. If a function is continuous at every value in an interval, then we say that the function is continuous in that interval.

Can a function be continuous only at rationals?

Continuous functions that are differentiable only at the rationals are shown to exist by Zahorski [6], but without an explicit construction. In this note, we give an explicit construction of a continuous function on [0, 1] that is differentiable only at the rationals in (0, 1).

What makes a function a rational function?

Introduction. A rational function is one that can be written as a polynomial divided by a polynomial. Since polynomials are defined everywhere, the domain of a rational function is the set of all numbers except the zeros of the denominator. f(x) = x / (x – 3). So the domain of f is the set of all numbers other than 3.

What is the integral value if f(x) = 1 and -1?

What is the integral value if f (x) =1 if x is rational, and f (x) =-1 if x is irrational (a=0,b=1)? You have to use Lebesgue integration to integrate that function. And the answer is -1. A characteristic (or indicator) function is one that is equal to one in a set, and zero otherwise.

What is the integral of the function of rationals?

Your function is a linear combination of the characteristic function of the rationals and a characteristic function of the irrationals on the interval . Since the rationals are a null set (have measure zero) your function has the same integral as if it were -1 everywhere on the interval .

What is the area under the function f(x) = 0?

The answer is 0, since the set of rational numbers has (Lebesgue) measure 0. If you flip the roles and make the function 0 at the rational numbers and 1 at the irrational numbers, the answer would be 1. Originally Answered: Why is the area under the function f (x) = 0 if x is rational and 1 otherwise one?

What is the value of Lebesgue’s integral of a function?

The answer below is for the original. The most natural way to interpret the question is to say that it’s asking about the integral of that function as ranges between 0 and 1. The answer is 0, since the set of rational numbers has (Lebesgue) measure 0.