MATHEMATICAL ANALYSIS 13

The Analysis View of Calculus

Analysis organises calculus into definitions, assumptions, theorems and logical consequences.

The logical structure

\[\boxed{\text{real numbers}\rightarrow\text{limits}\rightarrow\text{continuity}\rightarrow\text{derivatives and integrals}}\]

Limits

Limits make approximation precise and provide the definitions of derivatives and continuity.

Continuity

Continuity supports existence theorems such as the Intermediate and Extreme Value Theorems.

Differentiability

Differentiability gives local linear behaviour. Rolle's theorem and the Mean Value Theorem then connect local derivatives to behaviour across an interval.

Integration

Integration describes accumulation. The Fundamental Theorem of Calculus connects this global accumulation with local rates of change.

Approximation

Taylor's theorem uses derivatives to build polynomial approximations while retaining explicit information about error.

Infinite processes

Sequences and series are interpreted through limits. Convergence determines whether an infinite procedure represents a well-defined mathematical object.

The central habit of analysis

Do not use a theorem only because its formula looks appropriate. Check its hypotheses first.

\[\boxed{\text{What is assumed?}\quad\Longrightarrow\quad\text{What may be concluded?}}\]

This habit becomes increasingly important in differential equations, probability and stochastic calculus.