Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
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 , or with the Cauchy–Riemann equations
Total differentiability gives a local increment bound and therefore continuity
The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane