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statdad said: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 [tex] \rho [/tex] in the definition refers to the metric (distance function) defined for the range of the function [tex] f [/tex].
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!
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.
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.
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.
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.
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.