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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 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
The absolute value makes a metric space: is a metric, its open balls are the intervals , and it is unbounded
Zero sets and cozero sets of continuous real-valued functions