I with Real Analysis question

In summary, the conversation discusses the properties of boundedness and continuity for functions defined on the real numbers. It is given that f: R->R is continuous on all of R and B is a bounded subset of R. The first part of the conversation is about proving that the closure of B (cl(B)) is also a bounded set. The second part focuses on the image set f(B) and how it must also be a bounded subset of R. Lastly, the question is posed whether the function g:B->R, defined and continuous on B but not necessarily on all of R, must have a bounded image set g(B). This question requires either a proof or a counterexample.
  • #1
annastm
4
0
Suppose f: R->R is continuous on all of R and B is bounded subset of R.
a) show cl(B) is bounded set
b) show image set f(B) must be bounded subset of R
c) suppose g:B->R is defined & continuous on B but not necessarily on all of R - real #s, Must g(B) be bounded subset of R? (Prove or give counterexample)
 
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  • #2
Part of the 'rules' for this forum, that you accepted, is that you at least attempt to explain what effort you've made at doing the question, or where you think you need to start.
 

1. What is Real Analysis?

Real Analysis is a branch of mathematics that deals with the study of real numbers and functions. It involves the rigorous study of limits, continuity, differentiation, integration, and sequences and series.

2. What are the main applications of Real Analysis?

Real Analysis has a wide range of applications in various fields, such as physics, engineering, economics, and computer science. It is also used in advanced areas of mathematics, such as differential geometry and functional analysis.

3. What are the key concepts in Real Analysis?

Some of the key concepts in Real Analysis include limits, continuity, differentiation, integration, sequences and series, and convergence. These concepts are fundamental to understanding the behavior of real-valued functions.

4. How is Real Analysis different from Calculus?

Real Analysis is a more rigorous and abstract version of Calculus. While both deal with the study of functions and their behavior, Real Analysis focuses on the underlying principles and proofs, while Calculus focuses on computation and applications.

5. What are some useful resources for learning Real Analysis?

Some useful resources for learning Real Analysis include textbooks, online courses, and lecture notes from universities. It is also helpful to practice solving problems and proofs to gain a deeper understanding of the subject.

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