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A continuous function holomorphic off a single point is holomorphic

Statement

Let $U \subseteq C$ be open, let $p \in U$ , and let $f : U \to C$ be continuous on $U$ and holomorphic on $U ∖ {p}$ . Then $f$ is holomorphic on $U$ , the point $p$ included.

Facts & Assumptions

Given: An open set $U \subseteq C$ , a point $p \in U$ , and a function $f : U \to C$ that is continuous on $U$ and holomorphic on $U ∖ {p}$ .

[L1]

If $U \subseteq C$ is open, $p \in U$ , and $f : U \to C$ is continuous and holomorphic on $U ∖ {p}$ , then $\int_{\partial T} f (z) d z = 0$ for every filled triangle $T = Δ [a, b, c]$ of Filled complex triangles, their oriented three-edge boundary contours, diameter, and perimeter contained in $U$ ; the exceptional point may lie outside, inside, or on the boundary of $T$ (Goursat's triangle theorem remains valid for a continuous function holomorphic away from one point).

[L2]

If $Ω \subseteq C$ is open and $f : Ω \to C$ is continuous, then $f$ is holomorphic on $Ω$ if and only if $\int_{\partial Δ [a, b, c]} f (z) d z = 0$ whenever $Δ [a, b, c] \subseteq Ω$ ; repeated or collinear vertices are permitted (Morera's theorem: vanishing triangle integrals characterize holomorphy among continuous functions).

Proof

technique · direct

1.1

The hypotheses of [L1] are exactly the given ones, so $\int_{\partial T} f (z) d z = 0$ for every filled triangle $T \subseteq U$ , whether $p$ lies outside $T$ , inside it, or on its boundary.

givenL1

2.1

The function $f$ is continuous on the open set $U$ and step 1.1 supplies the vanishing triangle integrals demanded by the right-hand side of [L2], so [L2] makes $f$ holomorphic on all of $U$ , including at $p$ .

givenstep 1.1L2∎

Depends on

Used by

The filled difference quotient is holomorphic in each variable separately Lemma

Dependency tree · two levels

13 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

L. V. Ahlfors, Complex Analysis, 3rd ed., Ch. 4 §3.1 (standard reference, not scraped)