What Are Some Examples of Unique Real Analysis Functions?

In summary, real analysis is a branch of mathematics that focuses on the study of real numbers and their properties, with applications in various fields such as physics, economics, engineering, and computer science. Key concepts in real analysis include limits, continuity, differentiation, integration, sequences, and series. While closely related to calculus, real analysis is more theoretical and covers a broader range of topics. To improve understanding, a strong foundation in calculus and mathematical proofs is important, along with practicing and seeking guidance from knowledgeable sources.
  • #1
gordon_0451
1
0

Homework Statement


Please give examples

-functions continuous nowhere, continuous at one point

– functions differentiable everywhere but with discontinuous derivative

– Examples of uniformly continuous functions, functions not uniformly continuous

– Combinations of the above. For example, find a function, f : [0, 1] ! [0, 1], having the
intermediate-value property but continuous at only one point

thank you very much!


Homework Equations


Rolle’s theorem
Mean-value theorem
Cauchy mean-value theorem


The Attempt at a Solution



I have trouble to start it . Please help.
 
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  • #2
hi gordon_0451, welcome to PF - you need to show your attempts & people will try & help
 

FAQ: What Are Some Examples of Unique Real Analysis Functions?

What is real analysis?

Real analysis is a branch of mathematics that deals with the study of real numbers and their properties, such as continuity, limits, differentiation, and integration. It is a fundamental subject in mathematics that serves as the basis for many other areas, including calculus, differential equations, and topology.

What are some applications of real analysis?

Real analysis has numerous applications in various fields, including physics, economics, engineering, and computer science. For example, it is used in physics to model the motion of particles and in economics to study optimization problems. In engineering, real analysis is crucial in designing and analyzing systems with continuous variables, such as electrical circuits. In computer science, it is used for algorithms and data analysis.

What are the key concepts in real analysis?

Some of the key concepts in real analysis include limits, continuity, differentiation, integration, sequences, and series. These concepts are used to study the behavior and properties of real-valued functions and sets of numbers. They also form the foundation for more advanced topics, such as measure theory and functional analysis.

What are the differences between real analysis and calculus?

Real analysis and calculus are closely related, but there are some key differences between the two. Real analysis is a more rigorous and theoretical approach to studying real numbers and functions, while calculus is more focused on computation and applications. Real analysis also covers a broader range of topics, including sequences, series, and topology, while calculus focuses mainly on differentiation and integration.

How can I improve my understanding of real analysis?

To improve your understanding of real analysis, it is essential to have a strong foundation in calculus, linear algebra, and mathematical proofs. It is also helpful to practice solving problems and proofs, as well as reading and studying textbooks and other resources. Additionally, seeking guidance and assistance from a knowledgeable instructor or tutor can also greatly enhance your understanding of real analysis.

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