LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-17

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The filled difference quotient is continuous at its exceptional point and holomorphic away from it

Statement

Let $U \subseteq C$ be open, let $f : U \to C$ be holomorphic, and fix $z \in U$ . Define

$g (ζ) = {\begin{cases} \frac{f (ζ) - f (z)}{ζ - z}, & ζ \neq z, \\ f^{'} (z), & ζ = z . \end{cases}$

Then $g$ is continuous on $U$ and holomorphic on $U ∖ {z}$ . No holomorphy at the filled point $z$ is asserted.

Facts & Assumptions

Given: An open set $U$ , a holomorphic $f : U \to C$ , and a fixed point $z \in U$ .

[L1]

The derivative $f^{'} (z)$ is the limit of $(f (ζ) - f (z)) / (ζ - z)$ as $ζ \to z$ through $ζ \neq z$ (Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions).

[L2]

Sums, differences, and quotients with nonzero denominator of holomorphic functions are holomorphic (Linearity, product, reciprocal, and quotient rules for complex derivatives).

[L3]

A holomorphic function is continuous (Complex differentiability at a point implies continuity there).

Proof

technique · direct

1.1L1

By [L1], the off-point quotient tends to $f^{'} (z) = g (z)$ as $ζ \to z$ , which is exactly continuity of $g$ at $z$ ; this also covers constant $f$ .

1.2givenL2

On $U ∖ {z}$ , the numerator and denominator are holomorphic and the denominator is nonzero, so [L2] makes $g$ holomorphic there.

2.1step 1.1step 1.2L3∎

By [L3], step 1.2 also makes $g$ continuous away from $z$ ; together with step 1.1 this proves continuity on all of $U$ , without claiming differentiability at the filled point.

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lars Ahlfors, Complex Analysis, third edition, Ch. 4, Section 2.2 (standard reference, not scraped)