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Total differentiability gives a local increment bound and therefore continuity
Statement
If is totally differentiable at , then some satisfy whenever and . In particular is continuous at .
Facts & Assumptions
Given: A total derivative for at .
The normalized remainder in the total-derivative definition tends to zero as tends to zero (The total (Fréchet) derivative as the linear first-order approximation with remainder).
Every Euclidean linear map has a norm bound for some (Every Euclidean linear map has a unique matrix and satisfies for some ).
Proof
By [L1], choose such that the remainder satisfies whenever .
If bounds as in [L2], the triangle inequality gives for those , and it also holds at .
Given , take ; step 2.1 is the metric continuity condition at .
Depends on
- The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\|h\|_2)$ remainder
- Every Euclidean linear map has a unique matrix and satisfies $\|Lh\|_2\le K\|h\|_2$ for some $K\ge0$
- Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form
Used by
- Complex differentiability at a point implies continuity there Corollary
- The map y(x²+y²)/x off the line x=0, extended by zero on that line, has every directional derivative zero at the origin but is discontinuous there Counterexample
- Choice-free smooth inverse function theorem in Euclidean space Lemma
- Compatibility of smooth atlases is an equivalence relation, and smooth Euclidean maps compose Proposition
- Smooth maps are continuous Proposition
- On a convex open set, a uniform bound ‖Df(z)v‖₂≤ M‖v‖₂ implies ‖f(y)-f(x)‖₂≤ M‖y-x‖₂ Theorem
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a) Theorem
- The Euclidean inverse function theorem Theorem
Dependency tree · two levels
15 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, Basic Analysis I, §8.3 (standard reference, not scraped)