Recent content by alex.

The book is kinda old, so maybe they used different terms. But, hopefully micromass comment cleared the issue.

I am not sure I understand? I copied the question exactly like in the book.

By volume, it is okay to have volume of zero in which case our set ##f(x)>0## can be an empty set. By empty set I mean, has zero volume. Suppose A is empty. Fix any box I. Then I contains A. The volume is the integral of the function that equals 1 on A and 0 outside A. Thus the function is zero...

Homework Statement Let ##A\subset E^n## be a set with volume and ##f:A\to\mathbb{R}## a continuous function. Show that if the set ##\{x\in A:f(x)=0\}## has volume zero, then the set ##\{x\in A:f(x)>0\}## has volume. Homework Equations None The Attempt at a Solution A proposition...

1) Yes, it is a subset of ##\mathbb{R}.## 2) That's correct Hmm, maybe I am not sure what you mean? The phrase I used was the definition the book gave me for volume zero. Hmm, I should start by saying what exactly is ##\text{vol}(A)=\int_A 1,## the book didn't clarify? But to answer...

Homework Statement Let ##A\subset E^n## and let ##f:A\to E^m.## Consider the condition that there exist some ##M\in\mathbb{R}## such that ##d(f(x),f(y))\le Md(x,y)## for all ##x,y\in A.## Show that if the condition is satisfied, if ##m=n##, and ##\text{vol}(A)=0##, then...