Real Analysis — Sequences, Limits & Continuity

Mathematics Calculus & Analysis Cite
Primary: Weierstrass, Cauchy
Publication: Limits and continuity
Year: 19th c.
URL: Wikipedia

Description

Limit of sequence: a_n → L (ε-N definition). Limit of function: lim f(x) (ε-δ). Continuity at a point; uniform continuity. Bolzano-Weierstrass, Heine-Borel.

Dependency Flowchart

Note: Arrows mean "depends on". Assumes Chart 1 (completeness).

graph TD D1["Def: Sequence limit\na_n → L ε-N"] D2["Def: Function limit\nlim f ε-δ"] D3["Def: Continuity\nf cont at a"] D4["Def: Uniform continuity\nδ independent of x"] T1["Thm: Uniqueness\nof limit"] T2["Thm: BW\nbounded seq has conv subseq"] T3["Thm: Cont on [a,b]\n⇒ unif cont"] D1 --> D2 D2 --> D3 D3 --> D4 D1 --> T1 D1 --> T2 D3 --> T3 classDef definition fill:#b197fc,color:#fff,stroke:#9775fa classDef theorem fill:#51cf66,color:#fff,stroke:#40c057 class D1,D2,D3,D4 definition class T1,T2,T3 theorem

Color Scheme

Blue
Definitions
Teal
Theorems

Process Statistics

  • Nodes: 7
  • Edges: 8
  • Definitions: 4
  • Theorems: 3