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Complex differentiability at a point implies continuity there
Statement
If is complex differentiable at , then is continuous at .
Facts & Assumptions
Given: An open set , a point , and a function complex differentiable at .
Complex differentiability at is equivalent to real total differentiability there with derivative given by multiplication by a complex number (Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with , or with the Cauchy–Riemann equations).
If a Euclidean map is totally differentiable at a point, then it is continuous there (Total differentiability gives a local increment bound and therefore continuity).
Under , the modulus metric is exactly the Euclidean metric, and continuity on subsets of is metric continuity for this metric (The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane).
Proof
By [L1], is real totally differentiable at under the Euclidean identification.
By [L2], the coordinate map is continuous at ; [F1] identifies this with continuity in the complex modulus metric.
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
- FALSE: existence of partial derivatives satisfying Cauchy–Riemann everywhere on an open set implies holomorphy False statement
- A holomorphic function with zero derivative on a domain is constant Theorem
- Complex polynomials are entire with the power-rule derivative, and rational functions are holomorphic wherever their denominator is nonzero Theorem
- Linearity, product, reciprocal, and quotient rules for complex derivatives Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 63 results over 17 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- J. Lebl, Guide to Cultivating Complex Analysis, Proposition 2.1.2 (standard reference, not scraped)