Point values and continuity of real functions

Mathematical statement

For f:R→R, continuity at a means that every ε>0 admits δ>0 such that |f(x)−f(a)|<ε whenever |x−a|<δ. Equivalently, its punctured two-sided limit equals the defined f(a). In the two-affine-branch model with explicit center value V, continuity at a holds exactly when left limit=right limit=V.

Representative formula (exact TeX)

C(f,a)

Assumptions

• The whole real domain and center value are explicit; an undefined center is not automatically a discontinuity of a function declared on all R.
• Affine point-value questions compute ma+b. Branch questions compute left_m*a+left_b and right_m*a+right_b, with V supplied independently.
• For a continuous full-real extension of the removable quotient, the unique center value is 2a; before extension the original quotient remains defined only on R∖{a}.

Symbol scope

• C — continuity predicate at a point for a whole-real-domain function
• f — specified total real function or proposed extension
• a — specified real center

Intuition, definition, and misconceptions

Equal left and right limits alone are insufficient: they must also equal the center value. Agreement of the center with just one unequal side is insufficient as well.

An extension question asks how to define a new total real function, not whether the original punctured quotient already has a center value. Removability does not eliminate the need to define it.

The exact defect pair (left limit−V,right limit−V) shows the mismatch; both must be zero in this model. Common output translation, or slope changes preserving the same side limits, does not create a new defect calculation.

Common errors include ignoring the point value when one-sided limits agree, or claiming center continuity for an unextended quotient.

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 limits on punctured real domains — First check the two-sided limit.
• Function mapping — A center value is part of the total-function declaration.

Verification boundary

Declared branches and extension conditions can be recomputed; general continuity theory and formal Evidence have not been established.

not_formalized

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

Next obligation

Bind pinned ContinuousAt, the full real domain, and point extension, retaining the distinction from punctured limits and undefined points.

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.