# Continuous Functions

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

Topological Space - 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 ...
Arzelà–Ascoli Theorem - Generalizations - Compact Metric Spaces and Compact Hausdorff Spaces
... 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 ...
Mountain Climbing Problem - Formulation
... 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 ...
A Proof of Lusin's Theorem
... 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 ...
Hölder Condition - Examples
... 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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