Constant Continuity Adv. Calc 1

In summary, to prove that a function f: [a,b] ---> Q is continuous on [a,b], we can use the intermediate value theorem to show that f must be constant, since there is at least one irrational number between every two rational numbers. This eliminates the need for an epsilon-delta proof.
  • #1
chief12
11
0

Homework Statement


suppose f: [a,b] ---> Q is continuous on [a,b]. prove that f is constant on [a,b].

Homework Equations





The Attempt at a Solution



Since there is at least one irrational number between every two rational numbers,
then for f to be continuous in the given scenario, f must be constant

stuck about showing it with delta/epsilon proof
 
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  • #2
I don't think you need any epsilons and deltas. Just use the intermediate value theorem.
 
  • #3
Dick said:
I don't think you need any epsilons and deltas. Just use the intermediate value theorem.

can you explain more, a bit lost, test tomorrow
 
  • #4
chief12 said:
can you explain more, a bit lost, test tomorrow

Look up the intermediate value theorem and tell me what it says.
 

Related to Constant Continuity Adv. Calc 1

1. What is "Constant Continuity Adv. Calc 1"?

"Constant Continuity Adv. Calc 1" is a course in advanced calculus that focuses on the concept of continuity and its applications in mathematics and other scientific fields.

2. What topics are covered in "Constant Continuity Adv. Calc 1"?

The course covers topics such as limits, derivatives, integrals, and series, all with a focus on continuity. It also includes applications of continuity in areas such as optimization, physics, and engineering.

3. What are the prerequisites for taking "Constant Continuity Adv. Calc 1"?

Students are expected to have a strong foundation in calculus, including a thorough understanding of basic concepts such as limits, derivatives, and integrals. It is also recommended to have a solid understanding of algebra and trigonometry.

4. How is "Constant Continuity Adv. Calc 1" taught?

The course is typically taught through a combination of lectures, problem-solving sessions, and assignments. Some instructors may also incorporate group projects or real-world applications of continuity to enhance learning.

5. What are the career prospects for someone who has taken "Constant Continuity Adv. Calc 1"?

Students who have completed this course may pursue careers in fields such as mathematics, physics, engineering, computer science, and economics. The analytical and critical thinking skills gained from this course are highly valued in many industries and can lead to a wide range of opportunities.

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