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The function f(x) = √ x defined on [0, 1] is not Lipschitz continuous. This function becomes infinitely steep as x approaches 0 since its derivative becomes infinite. However, it is uniformly continuous, [8] and both Hölder continuous of class C 0, α for α ≤ 1/2 and also absolutely continuous on [0, 1] (both of which imply the former).
The definition of uniform continuity appears earlier in the work of Bolzano where he also proved that continuous functions on an open interval do not need to be uniformly continuous. In addition he also states that a continuous function on a closed interval is uniformly continuous, but he does not give a complete proof. [1]
3.1 Uniform, Hölder and Lipschitz continuity. 4 Continuous functions between topological spaces. ... A discontinuous function is a function that is not continuous.
There are examples of uniformly continuous functions that are not α –Hölder continuous for any α. For instance, the function defined on [0, 1/2] by f(0) = 0 and by f(x) = 1/log(x) otherwise is continuous, and therefore uniformly continuous by the Heine-Cantor theorem. It does not satisfy a Hölder condition of any order, however.
The Peano existence theorem shows only existence, not uniqueness, but it assumes only that f is continuous in y, instead of Lipschitz continuous. For example, the right-hand side of the equation dy / dt = y 1 / 3 with initial condition y(0) = 0 is continuous but not Lipschitz continuous.
Analogous results for better behaved classes of continuous functions do exist, for example the Lipschitz functions, whose set of non-differentiability points must be a Lebesgue null set (Rademacher's theorem). When we try to draw a general continuous function, we usually draw the graph of a function which is Lipschitz or otherwise well-behaved.
The following functions are absolutely continuous but not α-Hölder continuous: The function f(x) = x β on [0, c], for any 0 < β < α < 1; The following functions are absolutely continuous and α-Hölder continuous but not Lipschitz continuous: The function f(x) = √ x on [0, c], for α ≤ 1/2.
A sublinear modulus of continuity can easily be found for any uniformly continuous function which is a bounded perturbation of a Lipschitz function: if f is a uniformly continuous function with modulus of continuity ω, and g is a k Lipschitz function with uniform distance r from f, then f admits the sublinear module of continuity min{ω(t), 2r ...