Complex differentiability at a point implies continuity there

Depends on

• Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\bar z}f=0$, or with the Cauchy–Riemann equations
• Total differentiability gives a local $O(\|h\|_2)$ increment bound and therefore continuity
• The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane

Used by

• A holomorphic function equals its average on every circle inside a larger concentric holomorphy disc Corollary
• An injective holomorphic map has no critical point and is biholomorphic onto its image Corollary
• Cauchy's theorem for a null-homologous cycle Corollary
• Cauchy's theorem on a star-shaped domain: every closed rectifiable contour integral of a holomorphic function is zero Corollary
• Holomorphic functions are real analytic and smooth in their two real coordinates Corollary
• Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic Corollary
• Holomorphic integrals agree on homologous cycles Corollary
• Residues of p over q at a simple zero of q Corollary
• The higher-derivative form of the global Cauchy formula Corollary
• The principal logarithm is the normalised holomorphic branch on the slit plane Corollary
• A holomorphic function on an annulus can have a nonzero closed-contour integral Counterexample
• A nonvanishing holomorphic function on a domain with no holomorphic logarithm Counterexample
• Morera's theorem fails without continuity Counterexample
• Uniform convergence on the closed unit disc does not give a holomorphic extension to a larger disc Counterexample
• Integration over a complex chain and the index of a chain Definition
• The winding number of a closed contour about a point off its trace Definition
• Morera proves holomorphy of z↦∫₀¹ tᶻ dt on Rez>1 Example
• The three edge integrals of z² around the triangle with vertices 0, 1, and i sum to zero Example
• FALSE: every continuous complex-valued function on a domain has a primitive False statement
• FALSE: existence of partial derivatives satisfying Cauchy–Riemann everywhere on an open set implies holomorphy False statement
• A bounded separately holomorphic function on a polydisc is Lipschitz on every smaller polydisc Lemma
• A convergent complex power series with nonzero constant term has a convergent reciprocal power series locally Lemma
• A disc missing p carries a holomorphic logarithm of z-p Lemma
• At a simple pole the residue is the limit of (z-a)f(z) Lemma
• Dixon's glued function is entire and vanishes at infinity Lemma
• On a convex open set the difference quotient is an average of the derivative along the segment Lemma
• The Cauchy transform of a cycle is holomorphic off its trace, with the expected derivatives Lemma
• The filled difference quotient is continuous at its exceptional point and holomorphic away from it Lemma
• The filled difference quotient of a holomorphic function is jointly continuous Lemma
• The locally zero locus of a holomorphic function is clopen Lemma
• The logarithmic derivative has residue equal to local order Lemma
• A holomorphic function of several variables is continuous and separately holomorphic Proposition
• Star-shaped plane domains are homologically simply connected Proposition
• A holomorphic function equals its Taylor series throughout the largest centred disc in its domain Theorem
• A holomorphic function with zero derivative on a domain is constant Theorem
• A nonvanishing holomorphic function on a homologically simply connected domain has a holomorphic logarithm Theorem
• All higher complex derivatives exist and satisfy Cauchy's integral formula on an interior circle Theorem
• An absolutely convergent multi-indexed power series is holomorphic and differentiates termwise Theorem
• Cauchy's integral formula for a null-homologous cycle Theorem
• Characterizations of poles Theorem

…and 17 more results.

Dependency tree · two levels

19 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

• J. Lebl, Guide to Cultivating Complex Analysis, Proposition 2.1.2 (standard reference, not scraped)