Study ordered lists of real numbers.
Sequences provide a basic language for convergence and approximation.
Side 137
Real Analysis
Calculus rebuilt from definitions and proof: why limits work, when continuity is guaranteed, how convergence behaves and where familiar computational rules can fail.
sequence→limit→continuity→derivative→integral
04lenses
16working concepts
V0content
SS-1.0standard
Limits formalize controlled approximation.
The central question is whether outputs can be forced arbitrarily close by making inputs sufficiently close.
Express closeness without vague motion.
Quantifiers specify exactly how input tolerance controls output tolerance.
Detect convergence from internal behavior.
Completeness of the real numbers guarantees every Cauchy sequence converges.
Capture least upper bounds.
The supremum property is another expression of real-number completeness.
Continuity connects local limits to function values.
Continuous functions preserve small perturbations, but the precise consequences depend on domain structure.
Match the function value to the local limit.
Continuity is a local condition evaluated one point at a time.
Use one tolerance across the whole domain.
Uniform continuity is stronger and becomes important when domains are large or unbounded.
Continuous functions cannot jump over values.
This theorem turns topological connectedness into an existence result.
Continuous functions on compact sets attain maxima and minima.
Compactness supplies the global control needed for the guarantee.
Derivatives encode local linear approximation.
Differentiability is stronger than continuity and supports powerful local-to-global theorems.
Measure the limiting slope.
The derivative is a linear approximation rate, not merely a symbolic rule.
Connect local derivatives to total change.
The theorem underlies many error bounds and monotonicity arguments.
Approximate with polynomials.
Taylor formulas require conditions and remainder control; a smooth function need not equal its Taylor series.
Explore where intuition fails.
Nowhere-differentiable or discontinuous examples reveal why hypotheses matter.
Integration formalizes accumulation.
Different integration theories trade simplicity for generality and convergence power.
Approximate area with weighted partitions.
It works well for many bounded functions but has limitations under complex limiting operations.
Connect differentiation and integration.
Under appropriate conditions, accumulation and local rate become inverse operations.
Extend integration across infinite domains or singularities.
Convergence must be checked rather than assumed from symbolic antiderivatives.
Generalize what can be integrated.
Lebesgue theory reorganizes integration around measurable sets and convergence.
Analysis asks when intuition is actually justified. The familiar operations of calculus become theorems only after convergence, continuity and approximation are made precise.