# Real analysis definition question

• royzizzle
In summary, the function involving rho in this definition refers to the distance between the function f(x) and a particular value b in the range of the function. This definition is not specific to real functions of one variable and can also use other distance functions besides the absolute value.
royzizzle
what does the function involving rho mean in this definition?
(in picture)

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While the pictures demonstrate the notions of continuity and discontinuity at a real value, with graph in the plane, there is nothing in this definition that is specific to real functions of one variable. The $$\rho$$ in the definition refers to the metric (distance function) defined for the range of the function $$f$$.

While the pictures demonstrate the notions of continuity and discontinuity at a real value, with graph in the plane, there is nothing in this definition that is specific to real functions of one variable. The $$\rho$$ in the definition refers to the metric (distance function) defined for the range of the function $$f$$.

so rho(f(x),b) in this definition is the distance between f(x) and b in the range of f right?

k

thanks!

royzizzle said:
so rho(f(x),b) in this definition is the distance between f(x) and b in the range of f right?

k

thanks!

Yes. For real-valued functions as usually studied in calculus, both $$d$$ and $$\rho$$ are the absolute value, but in other situations other distance functions can be used.

## 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 analysis of functions, sequences, and series of real numbers, as well as their limits and continuity.

## 2. What are the key concepts in real analysis?

Some key concepts in real analysis include limits, continuity, differentiation, integration, sequences, series, and metric spaces. These concepts are used to understand and analyze the behavior of real-valued functions and their properties.

## 3. How is real analysis different from calculus?

Real analysis is a more rigorous and theoretical approach to the concepts studied in calculus. While calculus focuses on practical applications and computational methods, real analysis delves deeper into the foundations and properties of real numbers and functions.

## 4. What are some real-life applications of real analysis?

Real analysis has many real-life applications in fields such as physics, engineering, economics, and statistics. It is used to model and analyze real-world phenomena, such as the growth of populations, the spread of diseases, and the behavior of financial markets.

## 5. What skills are needed to study real analysis?

To study real analysis, one needs a strong foundation in calculus, algebra, and mathematical proofs. Critical thinking, problem-solving, and analytical skills are also important in understanding and applying the concepts of real analysis.

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