Real analysis is the rigorous mathematical study of real numbers, sequences, functions, and their fundamental properties. Unlike elementary calculus which focuses on computation, real analysis provides the theoretical foundation by proving why calculus works. The real numbers form a complete ordered field with no gaps, containing both rational numbers like fractions and irrational numbers like square root of 2 and pi.
A sequence is an ordered list of real numbers, like a₁, a₂, a₃, and so on. The fundamental concept is the limit of a sequence. We say the limit of sequence aₙ is L if the terms get arbitrarily close to L as n approaches infinity. This is formally defined using the epsilon-N definition: for every epsilon greater than zero, there exists a natural number N such that for all n greater than N, the absolute value of aₙ minus L is less than epsilon.
Function limits are defined using the epsilon-delta definition. The limit of f(x) as x approaches c equals L if for every epsilon greater than zero, there exists a delta greater than zero such that whenever x is within delta of c, f(x) is within epsilon of L. A function is continuous at point c if the limit as x approaches c equals f(c). This means there are no jumps or breaks in the function at that point.
The derivative of a function f at point c is defined as the limit of the difference quotient as h approaches zero. This represents the instantaneous rate of change of the function at that point, or geometrically, the slope of the tangent line to the curve at point c. As we make h smaller and smaller, the secant line through the two points approaches the tangent line, and the slope approaches the derivative.
The Riemann integral defines the area under a curve using Riemann sums. We partition the interval from a to b into n subintervals, each of width delta x equals b minus a over n. We form rectangles with heights determined by the function values, sum their areas, and take the limit as n approaches infinity. This process gives us the definite integral, representing the exact area under the curve between the limits of integration.