- #1

docnet

Gold Member

- 772

- 449

- Homework Statement
- I'm trying to come up with ways to find a continuous ##f## such that ##\sup_{||x||\leq 1}||f||=1## and ##f(x)\neq 1## with ##||x||=1##.

- Relevant Equations
- My understanding of this problem is a bit infantile. I want some math veterans to haze my poor understanding into shape.

Let ##f## be a continuous function defined in ##\mathbb{R}^n##. ##||\cdot ||## is the standard Euclidean metric. Then here are my suggested ways to choose ##f##:

$$1=\sup_{||x||\leq 1}||f||\neq \max_{||x||\leq 1}||f||$$ because the inequality ensures ##f(x)\neq 1## with ##||x||\leq 1##. Can you think of any specific examples?

Thank you.

edit:

Also, I'm wondering if category

**1.**Choose any continuous ##f## that satisfies$$1=\sup_{||x||\leq 1}||f||\neq \max_{||x||\leq 1}||f||$$ because the inequality ensures ##f(x)\neq 1## with ##||x||\leq 1##. Can you think of any specific examples?

**2.**Choose any continuous ##f## that satisfies $$1=\max_{||x||< 1}||f||$$ and $$1>\max_{||x||=1}f.$$ A simple example would be ##f=1-||x||##.Thank you.

edit:

**continuous**##f##Also, I'm wondering if category

**1**is invalid, i.e., if there do not exist functions that meet category**1.**
Last edited: