14.3 Lipschitz and Hölder functions[2DR]

Definition 358

[162] Let \(A⊂ ℝ\). A function \(f:A→ℝ \) is said Lipschitz continuous if there exists \(L{\gt}0\) such that \(∀ x,y∈ A\),

\[ |f(x)-f(y)|≤ L |x-y|~ ~ . \]

A function \(f:A→ℝ \) is said Hölder continuous if \(L{\gt}0\) and \(𝛼∈ (0,1]\) exist such that \(∀ x,y∈ A\),

\[ |f(x)-f(y)|≤ L |x-y|^𝛼~ ~ . \]

The constant \(𝛼\) is called the order.

As in the case of "uniform continuity", this notion extends to maps between metric spaces.

E358

[163]Prerequisites:1.Show that the Lipschitz functions, as well as Hölder functions, are uniformly continuous What can be said about their continuity modulus?

E358

[164]Let \(I⊂ ℝ\) be an open interval. Let \(f : I → ℝ\) be differentiable. Show that \(f'\) is bounded on \(I\), if and only if \(f\) is Lipschitz continuous.

E358

[165]Let \(I⊂ ℝ\) interval. Let \(f : I → ℝ\) such that there exists \(α {\gt} 1\) such that \(∀ x, y, |f (x) − f (y)| ≤ |x − y|^α\) (i.e. \(f\) is Hölder continuous of order \(𝛼{\gt}1\)): Show that f is constant.

E358

[166]Let be given \(f : [a, b] →ℝ\) and a decomposition of \([a, b]\) into intervals \(I_ 1 = [a, t_ 1 ], I_ 2 = [t_ 1 , t_ 2 ], \ldots , I_ n = [t_{n-1} , b]\) such that the restriction of \(f\) on each \(I_ k\) is Lipschitz of constant \(C\). Show that \(f\) is Lipschitz of constant \(C\).

Similarly for Hölder functions.

E358

[167]Let \(f : [a, b] →ℝ\) Hölder with exponent \(𝛼 ≤ 1\). Show that f is Hölderian with exponent \(𝛽\) for every \(𝛽 {\lt} 𝛼\).

Note that this is not technically true for \(f:ℝ→ℝ\).

E358

[169] Build \(f : [0, 1] →ℝ\) that is continuous but not Hölder continuous. Hidden solution: [UNACCESSIBLE UUID ’16B’][UNACCESSIBLE UUID ’16C’]

E358

[16D] A linear function \(f:ℝ^ n→ℝ^ k\) is Lipschitz.

E358

[16F]For each of the following functions, say if it is continuous, uniformly continuous, Hölder (and with which exponent), or Lipschitz.

  • \(f:(0,1)→ ℝ\), \(f(x)=\sin (1/x)\).

  • \(f:(0,1)→ ℝ\), \(f(x)=x^{1/x}\).

  • \(f:(1,∞)→ ℝ\), \(f(x)=\sin (x^ 2)/x\)

  • \(f:[-1,1]→ℝ\), \(f(x)=|x|^𝛽\) with \(𝛽{\gt}0\).

  • \(f:(0,∞)→ ℝ\), \(f(x)=\sin (x^𝛽)\) with \(𝛽{\gt}0\).

Hidden solution: [UNACCESSIBLE UUID ’16H’]

E358

[16J] Given \(L∈(0,1)\) if \(f:ℝ→ℝ\) satisfies

\[ |f(x)-f(y)|≤ L|x-y|\quad ∀ x,y∈ℝ \]

Then there is only one ”fixed point” that is a point \(x\) for which \(f(x)=x\).

E358

[16K] Find a function \(f:ℝ→ℝ\) such that

\[ |f(x)-f(y)|{\lt}|x-y|\quad ∀ x,y∈ℝ \]

but for which there is no ”fixed point” (that is a point \(x\) for which \(f(x)=x\)=.

Hidden solution: [UNACCESSIBLE UUID ’16M’]