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Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined
Statement
Let be continuous maps from a topological space. Then , , , , and are continuous. On the open cozero set , the quotient is continuous. The same holds for every finite sum, product, maximum, or minimum of continuous real-valued maps.
Facts & Assumptions
Given: A topological space and continuous maps .
A map into a product is continuous exactly when its coordinate maps are continuous, compositions of continuous maps are continuous, and a map whose range lies in a subspace is continuous into that subspace exactly when it is continuous into the ambient space (A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice, For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and , Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).
is open for continuous (Zero sets and cozero sets of continuous real-valued functions).
Proof
Addition is continuous at because gives . Multiplication is continuous there: after requiring , one has which is less than when both coordinate errors are smaller than . These coordinate conditions describe product neighbourhoods, so both operations are continuous.
The reverse triangle inequality makes absolute value continuous. Consequently are continuous by step 1.1 and composition.
Reciprocal is continuous at : if , then and Thus division is the product of and and is continuous on ; moreover is open by [F1].
The map is continuous by [L1], so composing it with the operations of steps 1.1 and 2.1 gives continuity of , , , and ; composing with absolute value gives continuity of .
Restricting and to and composing their product map with division gives continuity of there.
Iterating the binary operations of step 3.1 proves the finite assertions.
Depends on
- Continuity of a map of topological spaces at a point and globally
- The product set $\prod_{i \in I} X_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice
- For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and $f(\overline{A}) \subseteq \overline{f(A)}$
- The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$, and it is unbounded
- Zero sets and cozero sets of continuous real-valued functions
Used by
- A fibration need not be a locally trivial bundle Counterexample
- Without local finiteness, a pointwise finite sum of continuous functions can be discontinuous Counterexample
- A locally finite hat-function partition of unity on ℝ subordinate to overlapping intervals Example
- Brownian bridge from Brownian motion Example
- Endpoint-duplicating functions on [0,1] become all continuous functions on the endpoint quotient Example
- Hopf circle fibration Example
- Mobius band as an interval bundle with monodromy Example
- Radial second moment of multidimensional Brownian motion Example
- The inverse-square field is divergence free, and its flux through the sphere bounding the translated unit ball vanishes Example
- The polynomial algebra is dense but not closed on a nondegenerate compact interval Example
- Under choice and dependent choice, a finite subordinate partition of unity for a two-set cover of a compact interval Example
- A fixed-point-free self-map of the disk produces a continuous retraction onto the unit circle Lemma
- A locally finite family of continuous nonnegative functions has a continuous pointwise sum Lemma
- A locally finite nonnegative family with positive pointwise sum normalizes to a partition of unity Lemma
- A root-free complex polynomial gives nullhomotopic normalized circle loops Lemma
- Antipodal complements cover Sⁿ by simply connected sets with path-connected overlap for n≥2 Lemma
- Finite join models for the circle and the two-point group Lemma
- Pointwise multiplication and concatenation of loops in a topological group agree up to homotopy Lemma
- The filled difference quotient of a holomorphic function is jointly continuous Lemma
- The normalized large-radius loop of a monic degree-n polynomial has degree n Lemma
- Numerable fiber bundles are hurewicz fibrations Theorem
- The complex exponential is entire and its complex derivative is itself Theorem
- The disc formula for the volume of a solid of revolution Theorem
Dependency tree · two levels
39 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. Robbin, Partitions of Unity (standard reference, not scraped)
- S. Semmes, Topology notes, Sections 5.13–5.14 (Rice University) (standard reference, not scraped)
- Continuity notes (University of California, Berkeley) (standard reference, not scraped)