Continuous Functions

Checkpoints

• Continuous Function Basis A function is continuous at a point if its limit there equals its value there. This checkpoint defines continuity at a point and on a set, proves the algebra of continuous functions, and establishes that every elementary function is continuous on its domain.
• Extreme Value Theorem Basis A continuous function on a closed bounded interval attains both a maximum and a minimum. This checkpoint proves the extreme value theorem using boundedness arguments, and explains why dropping any hypothesis — continuity, closedness, or boundedness — destroys the conclusion.
• Intermediate Value Theorem Basis If f is continuous on [a, b] and y lies between f(a) and f(b), then there exists c ∈ [a, b] with f(c) = y. This checkpoint proves the theorem using the completeness of ℝ, and applies it to root-finding and existence proofs.
• Limit of a Function Basis The limit of a function describes the value f(x) approaches as x approaches a point. This checkpoint gives both the ε–δ and sequential characterisations, proves they are equivalent, and uses them to compute limits of typical real functions.
• Local Properties of Limits Basis Limits behave well with arithmetic and order locally: if f and g have limits at a point, so do f±g, fg, and f/g (when the denominator is nonzero), and inequalities pass to the limit. This checkpoint proves these arithmetic and order theorems, the squeeze theorem, and the local sign-preservation property.