Function limits on punctured real domains

Mathematical statement

For a real function f defined at all x≠a sufficiently near a, lim_(x→a)f(x)=L means that each ε>0 admits δ>0 with |f(x)−L|<ε whenever 0<|x−a|<δ. This pulse uses affine expressions, removable-factor quotients, and left/right affine branches; a finite two-sided limit requires equal one-sided limits.

Representative formula (exact TeX)

L(f,a)

Assumptions

• Exercise functions use R∖{a}, with no requirement that f(a) exist; isolated points and other unspecified approach domains are excluded.
• The affine limit of mx+b is ma+b. Cancel x−a in (x²−a²)/(x−a) only at x≠a; the limit is 2a, without assigning the original quotient a center value.
• For positive rational ε=p/q and δ=u/v, the uniform affine bound permits |m|u q≤p v. Equality is allowed because input distance is strictly below δ; at m=0 every positive δ works.

Symbol scope

• L — punctured real-limit operation
• f — specified function defined away from the center
• a — real approach point

Intuition, definition, and misconceptions

A punctured limit excludes the center. An undefined or changed center value therefore does not directly change the limit, unlike continuity, which requires a center value.

The affine bound follows from |f(x)−(ma+b)|=|m||x−a|. Strict input distance allows |m|δ=ε while retaining strict output error; do not copy the strict cutoff comparison from the sequence model.

The affine branches have their own one-sided limits. Unequal values defeat the finite two-sided limit without erasing either legitimate one-sided limit. Finite samples do not establish general limit claims.

Common errors include identifying a limit with f(a), filling a center value by division by zero, or rejecting a valid δ at equality.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Function mapping — Specify domain and codomain.
• Tends-to relation — Approaching a point is not direct substitution.
• Absolute value — Use real distance and ε–δ error.
• Polynomial factorization over the rationals — Cancel x−a only with a nonzero denominator.

Verification boundary

Explicit affine, removable-quotient, and two-sided models can be checked exactly; general function-limit and type bridges have no attached formal Evidence.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind punctured nhds, one-sided neighborhoods, and affine error bounds; totalized division at zero does not replace punctured-domain eligibility.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.