StudyGPT Just Leveled Up – Say Hello to Knowee! 🚀
Knowee
Questions
Features
Study Tools

Consider the function f : R → R defined by f (x)=1 if x∈ Q, f (x)=0 if x∈ R/Q, where is f continuous? be sure to prove your assertion

Question

🧐 Not the exact question you are looking for?Go ask a question

Solution

The function you've described is known as the Dirichlet function. It's a classic example of a function that is discontinuous at every point in its domain.

Here's a step-by-step proof:

  1. Let's first understand the function. The function f(x) is defined to be 1 for all rational numbers (numbers that can be expressed as a quotient of two integers) and 0 for all irrational numbers (numbers that cannot be expressed as a quotient of two integers).

  2. We want to show that this function is discontinuous at every point in its domain. To do this, we'll use the definition of continuity. A function f(x) is continuous at a point x=a if the limit as x approaches a of f(x) equals f(a).

  3. Let's consider an arbitrary point a in the domain of f. Without loss of generality, we can consider two cases: a is rational and a is irrational.

  4. If a is rational, then f(a) = 1. However, for any neighborhood around a, no matter how small, there will always be irrational numbers. For these numbers, f(x) = 0. Therefore, the limit as x approaches a of f(x) does not exist, so f is not continuous at a.

  5. If a is irrational, then f(a) = 0. However, for any neighborhood around a, no matter how small, there will always be rational numbers. For these numbers, f(x) = 1. Therefore, the limit as x approaches a of f(x) does not exist, so f is not continuous at a.

  6. Since a was an arbitrary point in the domain of f, we can conclude that f is discontinuous at every point in its domain.

This problem has been solved

Upgrade your grade with Knowee

Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.