Basic Analysis I — Real Analysis · Problem 3.4.13

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · uniform-continuity

Problem

a) If

f : [0, 1] \to \mathbb{R}

is given by

f(x) := x^m

for an integer

m \geq 0

, show

f

is Lipschitz and find the best (the smallest) Lipschitz constant

K

(depending on

m

of course). Hint:

(x - y)(x^{m-1} + x^{m-2}y + x^{m-3}y^2 + \cdots + xy^{m-2} + y^{m-1}) = x^m - y^m

. b) Using the previous exercise, show that if

f : [0, 1] \to \mathbb{R}

is a polynomial, that is,

f(x) := a_m x^m + a_{m-1}x^{m-1} + \cdots + a_0

, then

f

is Lipschitz.