Question

The goal of long questions is to ask students familiarize and understand that mathematical solutions are built step by step. To attempt these questions, one has to read every part of the questions very carefully. Each part will lead to the answer of the next part. By doing these follow-up exercises, students can learn to attempt questions step by step. LaTex typed solution is preferred in this exercise. Q1 ($\alpha$-Holder functions) Let $\alpha > 0$. A real-valued function $f: [a, b] \rightarrow \mathbb{R}$ is called a $\alpha$-Holder function if there exists a constant $C > 0$ such that $|f(x) - f(y)| \leq C|x - y|^\alpha$ for all $x, y \in [a, b]$. If $\alpha = 1$, $f$ is called a Lipschitz function (e.g. $f(x) = x$ is Lipschitz). (a) Show that a $\alpha$-Holder function must be uniformly continuous. (b) Show that if $\alpha > 1$, then $f = 0$. (c) Using Intermediate value theorem, show that if $f: [0, 1] \rightarrow [0, 1]$ is a continuous function, then there exists a fixed point $x_0$ such that $f(x_0) = x_0$. (d) Show that if $f: [0, 1] \rightarrow [0, 1]$ is a continuous function and $f$ is Lipschitz with constant $C < 1$, then the fixed point found in (c) is unique. (Remark: (d) is a special case of the Banach fixed point theorem)

          The goal of long questions is to ask students familiarize and understand that
mathematical solutions are built step by step. To attempt these questions, one has
to read every part of the questions very carefully. Each part will lead to the answer
of the next part. By doing these follow-up exercises, students can learn to attempt
questions step by step.
LaTex typed solution is preferred in this exercise.
Q1 ($\alpha$-Holder functions)
Let $\alpha > 0$. A real-valued function $f: [a, b] \rightarrow \mathbb{R}$ is called a $\alpha$-Holder function if
there exists a constant $C > 0$ such that
$|f(x) - f(y)| \leq C|x - y|^\alpha$
for all $x, y \in [a, b]$. If $\alpha = 1$, $f$ is called a Lipschitz function (e.g. $f(x) = x$ is
Lipschitz).
(a) Show that a $\alpha$-Holder function must be uniformly continuous.
(b) Show that if $\alpha > 1$, then $f = 0$.
(c) Using Intermediate value theorem, show that if $f: [0, 1] \rightarrow [0, 1]$ is a continuous
function, then there exists a fixed point $x_0$ such that $f(x_0) = x_0$.
(d) Show that if $f: [0, 1] \rightarrow [0, 1]$ is a continuous function and $f$ is Lipschitz with
constant $C < 1$, then the fixed point found in (c) is unique.
(Remark: (d) is a special case of the Banach fixed point theorem)

The goal of long questions is to ask students familiarize and understand that
mathematical solutions are built step by step. To attempt these questions, one has
to read every part of the questions very carefully. Each part will lead to the answer
of the next part. By doing these follow-up exercises, students can learn to attempt
questions step by step.
LaTex typed solution is preferred in this exercise.
Q1 (α-Holder functions)
Let α > 0. A real-valued function f: [a, b] →ℝ is called a α-Holder function if
there exists a constant C > 0 such that
|f(x) - f(y)| ≤ C|x - y|^α
for all x, y ∈ [a, b]. If α = 1, f is called a Lipschitz function (e.g. f(x) = x is
Lipschitz).
(a) Show that a α-Holder function must be uniformly continuous.
(b) Show that if α > 1, then f = 0.
(c) Using Intermediate value theorem, show that if f: [0, 1] → [0, 1] is a continuous
function, then there exists a fixed point x0 such that f(x0) = x0.
(d) Show that if f: [0, 1] → [0, 1] is a continuous function and f is Lipschitz with
constant C < 1, then the fixed point found in (c) is unique.
(Remark: (d) is a special case of the Banach fixed point theorem)

Added by Richard S.

Question

Please give Ace some feedback