# General question regarding continuous functions and spaces

• TwilightTulip
In summary, the continuity of h(x) depends on the continuity of f and g on their respective domains. It is not necessarily true that h(x) will be continuous if f and g agree on the closure or boundary of A. There are some results that address the extension of continuous functions, such as the Tietze Extension and the extension of uniformly-continuous functions on the rationals to the reals. However, extending homeomorphisms from a subspace to a superspace can be more challenging and requires certain conditions to be satisfied. Ultimately, the continuity of h(x) is determined by the continuity of f and g on their respective domains.
TwilightTulip
Let X be some topological space. Let A be a subspace of X. I am thinking about the following: If f is a cts function from X to X, and g a cts function from X to A, when is the piece-wise function

h(x) = f(x) if x is not in A, g(x) if x is in A

continuous? My intuition tells me they must agree on the closure (or maybe boundary?) of A? If not, any idea?

This is a pretty broad question (still, I like pretty broads). Some results:

Tietze Extension: If A is a closed subset of a normal space X, and f:A-->R is continuous, then f extends continuously to X (this is constructive result, i.e., there is a method to construct the extension).

If f is a uniformly-continuous function on the rationals, then f extends to the reals. Use the fact that uniformly-continuous functions preserve Cauchy-sequences, and use the continuity condition: f continuous ( in metric or 1st-countable space) iff [(x_n-->x)->(f(x_n)-->f(x))] ; left-right is true in any topological space (use nets instead of sequences for general non-metrizable). Note that continuity alone is not enough: use, e.g: f(x)=1/(x-sqr2), or 1/(x-pi) from rationals to rationals, it is continuous, but does not extend .

There are also results on functions on manifolds defined on individual charts, that can be put together into globally-defined functions, using partitions of unity. This uses the fact that manifolds are paracompact (I think Hausdorff +2nd-countable => paracompact). If you want more detail, let me know, because I need to go to sleep soon.

Harder stil, is the extension of homeomorphisms from a subspace into the superspace. In the case of S^4, if the subspace is trivially-embedded (i.e., unknotted, or any two embeddings are isotopic to each other) , then the maps that extend are precisely those that preserve a quadratic form called the Rohklin form.

It is not necessarily true that h(x) will be continuous if f and g agree on the closure or boundary of A. The continuity of h(x) depends on the continuity of f and g on their respective domains.

For example, if f is continuous on X but g is not continuous on A, then h(x) will not be continuous. This is because h(x) will not be continuous at points in A where g is not continuous.

On the other hand, if both f and g are continuous on their respective domains, then h(x) will be continuous. This is because both f and g are continuous on their domains and h(x) is simply a combination of the two functions.

Therefore, the continuity of h(x) depends on the continuity of f and g on their respective domains, and not necessarily on the closure or boundary of A.

## What is a continuous function?

A continuous function is a type of mathematical function that has no sudden jumps or breaks in its graph. This means that the function can be drawn without lifting the pencil from the paper.

## What is the difference between a continuous and a discontinuous function?

A continuous function is one where there are no sudden jumps or breaks in its graph, while a discontinuous function has at least one point where the graph has a sudden jump or break.

## What is a continuous space?

A continuous space is a mathematical concept that refers to a space that has no gaps or discontinuities. This means that every point in the space can be reached by continuously moving through the space without any sudden jumps or breaks.

## What is the importance of continuity in mathematics?

Continuity is an important concept in mathematics because it allows us to understand and describe how objects and functions behave in a smooth and predictable manner. This is essential in many branches of mathematics, such as calculus and topology.

## What are some real-world applications of continuous functions and spaces?

Continuous functions and spaces have numerous applications in the real world, including in physics, engineering, and economics. They are used to model and predict the behavior of systems and phenomena, such as motion, electricity, and market trends.

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