### Some articles on *continuous functions, function, continuous, functions*:

**Continuous Functions**

... A

**function**f X→ Y between topological spaces is called

**continuous**if for all x ∈ X and all neighbourhoods N of f(x) there is a neighbourhood M of x such that f(M) ⊆ N ... Equivalently, f is

**continuous**if the inverse image of every open set is open ... that there are no "jumps" or "separations" in the

**function**...

... let C(X) be the space of real-valued

**continuous functions**on X ... if for every x ∈ X, A version of this holds also in the space C(X) of real-valued

**continuous functions**on a compact Hausdorff space X (Dunford Schwartz 1958, §IV.6.7) Let X be a compact Hausdorff ... is thus a fundamental result in the study of the algebra of

**continuous functions**on a compact Hausdorff space ...

... The following result is due to Huneke Suppose and are

**continuous functions**from to with and, and such that neither

**function**is constant on an interval ... Then there exist

**continuous functions**and from to with, and such that, where "" stands for a composition of

**functions**... To see heuristically that the result does not extend to all

**continuous functions**, note that if has a constant interval while has a highly oscillating interval on the same level, then the first ...

... Since

**continuous functions**are dense in L1, there exists a sequence of

**continuous functions**gn tending to f in the L1 norm ... Since uniform limits of

**continuous functions**are

**continuous**, the theorem is proved ...

... If 0 < α ≤ β ≤ 1 then all Hölder

**continuous functions**on a bounded set Ω are also Hölder

**continuous**... This also includes β = 1 and therefore all Lipschitz

**continuous functions**on a bounded set are also C0, α Hölder

**continuous**... The

**function**defined on is not Lipschitz

**continuous**, but is C0, α Hölder

**continuous**for α ≤ 1/2 ...

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