Can the Triangle Inequality Be Applied to Functions?

Thread starter ares25
Start date Apr 17, 2014
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Analysis Intro Real analysis

In summary, real analysis is a branch of mathematics that focuses on the study of real numbers, their properties, and their behavior. It is important because it provides a foundation for other areas of math and has practical applications in fields such as physics and economics. Key concepts in real analysis include limits, continuity, differentiability, and integration. It is more rigorous and abstract than calculus, which focuses more on problem-solving techniques. Resources for learning real analysis include textbooks, online courses, video lectures, and practicing problem-solving and proofs.

Apr 17, 2014

ares25

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Sorry for the post. I'll take it down soon.
Thanks for the help

Last edited: Apr 17, 2014

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Apr 17, 2014

jbunniii

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Use the triangle inequality in this form:
$$||u|-|v|| \leq |u-v|$$
with ##u = f(x) - x## and ##v = f(y) - y##.

1. What is real analysis?

Real analysis is a branch of mathematics that deals with the study of real numbers and their properties. It involves the use of rigorous mathematical proofs to understand the behavior and structure of real numbers, functions, and sequences.

2. Why is real analysis important?

Real analysis is important because it provides a foundation for other branches of mathematics, such as calculus and differential equations. It also has applications in various fields, including physics, engineering, and economics.

3. What are some key concepts in real analysis?

Some key concepts in real analysis include limits, continuity, differentiability, and integration. These concepts help to understand the behavior and properties of functions and sequences.

4. How is real analysis different from calculus?

Real analysis is more abstract and rigorous than calculus. It focuses on the properties of real numbers and functions, while calculus is more concerned with techniques for solving problems involving real numbers and functions.

5. What are some resources for learning real analysis?

Some resources for learning real analysis include textbooks, online courses, and video lectures. It is also helpful to practice solving problems and working through proofs to gain a deeper understanding of the concepts.