024 M 4159213 Section - 1Select The Correct Answer. Options:Let $f$ Be The Function Defined On $[0,1]$ By $f(x) = -1$ When $x$ Is Irrational.A. $f \in R [a, B]$B. $\left| \int \right| \in

Mar 8, 2025 by ADMIN 201 views

024 M 4159213 Section - 1

Select the Correct Answer

Function Defined on a Closed Interval

Let $f$ be the function defined on $[0,1]$ by $f(x) = -1$ when $x$ is irrational. This function is a classic example of a discontinuous function, as it has a different value for rational and irrational numbers.

Rational and Irrational Numbers

Rational numbers are those that can be expressed as the ratio of two integers, such as $\frac{1}{2}$ or $\frac{3}{4}$. Irrational numbers, on the other hand, are those that cannot be expressed as a ratio of integers, such as $\pi$ or $\sqrt{2}$.

Discontinuity of the Function

The function $f(x) = -1$ when $x$ is irrational is discontinuous because it has a different value for rational and irrational numbers. This means that the function does not have a limit at any point in the interval $[0,1]$.

Riemann Integrability

A function is Riemann integrable if it is bounded and has at most a countable number of discontinuities in the interval. In this case, the function $f(x) = -1$ when $x$ is irrational is bounded, but it has an uncountable number of discontinuities in the interval $[0,1]$.

Correct Answer

Based on the above discussion, we can conclude that the function $f(x) = -1$ when $x$ is irrational is not Riemann integrable on the interval $[0,1]$.

A. $f \in R [a, b]$

This option is incorrect because the function $f(x) = -1$ when $x$ is irrational is not Riemann integrable on the interval $[0,1]$.

B. $\left| \int \right| \in R [a, b]$

This option is also incorrect because the function $f(x) = -1$ when $x$ is irrational is not Riemann integrable on the interval $[0,1]$.

C. $f \notin R [a, b]$

This option is correct because the function $f(x) = -1$ when $x$ is irrational is not Riemann integrable on the interval $[0,1]$.

D. $\left| \int \right| \notin R [a, b]$

This option is also correct because the function $f(x) = -1$ when $x$ is irrational is not Riemann integrable on the interval $[0,1]$.

Conclusion

In conclusion, the correct answer is C. $f \notin R [a, b]$, which means that the function $f(x) = -1$ when $x$ is irrational is not Riemann integrable on the interval $[0,1]$.

Riemann Integrability Theorem

The Riemann integrability theorem states that a function is Riemann integrable if and only if it is bounded and has at most a countable number of discontinuities in the interval.

Bounded Function

A function is bounded if it has a finite upper bound and a finite lower bound. In this case, the function $f(x) = -1$ when $x$ is irrational is bounded because it has a finite upper bound of $-1$ and a finite lower bound of $-1$.

Countable Number of Discontinuities

A function has at most a countable number of discontinuities if it has a finite or countably infinite number of discontinuities in the interval. In this case, the function $f(x) = -1$ when $x$ is irrational has an uncountable number of discontinuities in the interval $[0,1]$.

Riemann Integrability Theorem Conclusion

Based on the Riemann integrability theorem, we can conclude that the function $f(x) = -1$ when $x$ is irrational is not Riemann integrable on the interval $[0,1]$.

Riemann Sum

The Riemann sum is a mathematical concept used to approximate the value of a definite integral. It is defined as the sum of the areas of the rectangles that approximate the area under the curve.

Riemann Sum Formula

The Riemann sum formula is given by:

\sum_{i=1}^{n} f(x_i) \Delta x

where $f(x_i)$ is the value of the function at the point $x_i$, and $\Delta x$ is the width of the rectangle.

Riemann Sum Conclusion

Based on the Riemann sum formula, we can conclude that the Riemann sum is a mathematical concept used to approximate the value of a definite integral.

Definite Integral

The definite integral is a mathematical concept used to find the area under a curve. It is defined as the limit of the Riemann sum as the number of rectangles approaches infinity.

Definite Integral Formula

The definite integral formula is given by:

\int_{a}^{b} f(x) dx

where $f(x)$ is the function, and $a$ and $b$ are the limits of integration.

Definite Integral Conclusion

Based on the definite integral formula, we can conclude that the definite integral is a mathematical concept used to find the area under a curve.

Conclusion

In conclusion, the function $f(x) = -1$ when $x$ is irrational is not Riemann integrable on the interval $[0,1]$ because it has an uncountable number of discontinuities in the interval. The Riemann sum is a mathematical concept used to approximate the value of a definite integral, and the definite integral is a mathematical concept used to find the area under a curve.

References

[1] Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
[2] Bartle, R. G. (1976). The Elements of Real Analysis. Wiley.
[3] Rudin, W. (1987). Real and Complex Analysis. McGraw-Hill.

Frequently Asked Questions

Q1: What is the function defined on [0,1] by f(x) = -1 when x is irrational?

A1: The function is defined as f(x) = -1 when x is irrational, and it is a classic example of a discontinuous function.

Q2: Why is the function f(x) = -1 when x is irrational not Riemann integrable on the interval [0,1]?

A2: The function is not Riemann integrable on the interval [0,1] because it has an uncountable number of discontinuities in the interval.

Q3: What is the Riemann integrability theorem?

A3: The Riemann integrability theorem states that a function is Riemann integrable if and only if it is bounded and has at most a countable number of discontinuities in the interval.

Q4: What is the Riemann sum?

A4: The Riemann sum is a mathematical concept used to approximate the value of a definite integral. It is defined as the sum of the areas of the rectangles that approximate the area under the curve.

Q5: What is the definite integral?

A5: The definite integral is a mathematical concept used to find the area under a curve. It is defined as the limit of the Riemann sum as the number of rectangles approaches infinity.

Q6: Why is the function f(x) = -1 when x is irrational not bounded?

A6: The function is not bounded because it has a finite upper bound of -1 and a finite lower bound of -1, but it is not bounded in the sense that it does not have a finite upper bound and a finite lower bound.

Q7: What is the difference between a bounded function and a function with a finite upper bound and a finite lower bound?

A7: A bounded function is a function that has a finite upper bound and a finite lower bound, whereas a function with a finite upper bound and a finite lower bound is a function that has a finite upper bound and a finite lower bound, but it may not be bounded.

Q8: Why is the function f(x) = -1 when x is irrational not Riemann integrable on the interval [0,1] because it has an uncountable number of discontinuities in the interval?

A8: The function is not Riemann integrable on the interval [0,1] because it has an uncountable number of discontinuities in the interval, which means that it does not have a limit at any point in the interval.

Q9: What is the difference between a countable number of discontinuities and an uncountable number of discontinuities?

A9: A countable number of discontinuities is a finite or countably infinite number of discontinuities, whereas an uncountable number of discontinuities is an uncountably infinite number of discontinuities.

Q10: Why is the function f(x) = -1 when x is irrational not Riemann integrable on the interval [0,1] because it has an uncountable number of discontinuities in the interval?

A10: The function is not Riemann integrable on the interval [0,1] because it has an uncountable number of discontinuities in the interval, which means that it does not have a limit at any point in the interval.

Conclusion

In conclusion, the function f(x) = -1 when x is irrational is not Riemann integrable on the interval [0,1] because it has an uncountable number of discontinuities in the interval. The Riemann sum is a mathematical concept used to approximate the value of a definite integral, and the definite integral is a mathematical concept used to find the area under a curve.

References

[1] Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
[2] Bartle, R. G. (1976). The Elements of Real Analysis. Wiley.
[3] Rudin, W. (1987). Real and Complex Analysis. McGraw-Hill.

024 M 4159213 Section - 1Select The Correct Answer. Options:Let $f$ Be The Function Defined On $[0,1]$ By $f(x) = -1$ When $x$ Is Irrational.A. $f \in R [a, B]$B. $\left| \int \right| \in

Select the Correct Answer

Function Defined on a Closed Interval

Rational and Irrational Numbers

Discontinuity of the Function

Riemann Integrability

Correct Answer

A. $f \in R [a, b]$

B. $\left| \int \right| \in R [a, b]$

C. $f \notin R [a, b]$

D. $\left| \int \right| \notin R [a, b]$

Conclusion

Riemann Integrability Theorem

Bounded Function

Countable Number of Discontinuities

Riemann Integrability Theorem Conclusion

Riemann Sum

Riemann Sum Formula

Riemann Sum Conclusion

Definite Integral

Definite Integral Formula

Definite Integral Conclusion

Conclusion

References

Further Reading

Frequently Asked Questions

Q1: What is the function defined on [0,1] by f(x) = -1 when x is irrational?

Q2: Why is the function f(x) = -1 when x is irrational not Riemann integrable on the interval [0,1]?

Q3: What is the Riemann integrability theorem?

Q4: What is the Riemann sum?

Q5: What is the definite integral?

Q6: Why is the function f(x) = -1 when x is irrational not bounded?

Q7: What is the difference between a bounded function and a function with a finite upper bound and a finite lower bound?

Q8: Why is the function f(x) = -1 when x is irrational not Riemann integrable on the interval [0,1] because it has an uncountable number of discontinuities in the interval?

Q9: What is the difference between a countable number of discontinuities and an uncountable number of discontinuities?

Q10: Why is the function f(x) = -1 when x is irrational not Riemann integrable on the interval [0,1] because it has an uncountable number of discontinuities in the interval?

Conclusion

References

Further Reading