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.