Are there any functions which are uniform continuous and not Lipschitz continuous.
Dear followers,
I will first summarize your answers.
A real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, α, such that
| f(x) - f(y) || \leq C |x - y ||^{\alpha}
for all x and y in the domain of f. More generally, the condition can be formulated for functions between any two metric spaces. The number α is called the exponent of the Hölder condition. If α = 1, then the function satisfies a Lipschitz condition
If function is Lipschitz continuous, then it is Holder continuous
and
all Holder continuous function are uniformly continuous.
Thus,
we have the following chain of inclusions for functions over a compact subset of the real line
Continuously differentiable ⊆ Lipschitz continuous ⊆ α-Hölder continuous ⊆ uniformly continuous ⊆ continuous
where 0 < α ≤1.
For example,
f(x) =x1/2 is uniformly continuous but not Lipschitz continous on the interval (0, 1). Here the derivative f'(x) =1/2 x-1/2 is not bounded in nbg 0.
All Libschitz continuous function are uniformly continuous, but the converse is not true.
The relation between libschitz and uniform continuity is as follows
Continuously differentiable ⊆ Lipschitz continuous ⊆ α-Hölder continuous ⊆ uniformly continuous ⊆ continuous
where 0
Definition. A function f defined on a set S⊆R is said to be Lipschitz continuous on S if there exists an M so that|f(x)−f(c)||x−c|≤M
for all x and c in S such that x≠c.
My Heuristic Interpretation: if f
is Lipschitz continuous then the "absolute slope" of fis never unbounded i.e. no asymptotes.
Definition. A continuous function f
defined on Dom(f) is said to be uniformly continuous if for each ε>0 ∃ δ>0 s.t. ∀ x,c∈Dom(f)|x−c|≤δ ⇒ |f(x)−f(c)|≤ε
Dear followers,
I will first summarize your answers.
A real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, α, such that
| f(x) - f(y) || \leq C |x - y ||^{\alpha}
for all x and y in the domain of f. More generally, the condition can be formulated for functions between any two metric spaces. The number α is called the exponent of the Hölder condition. If α = 1, then the function satisfies a Lipschitz condition
If function is Lipschitz continuous, then it is Holder continuous
and
all Holder continuous function are uniformly continuous.
Thus,
we have the following chain of inclusions for functions over a compact subset of the real line
Continuously differentiable ⊆ Lipschitz continuous ⊆ α-Hölder continuous ⊆ uniformly continuous ⊆ continuous
where 0 < α ≤1.
For example,
f(x) =x1/2 is uniformly continuous but not Lipschitz continous on the interval (0, 1). Here the derivative f'(x) =1/2 x-1/2 is not bounded in nbg 0.
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