A set A of real numbers is said to be dense if every open...

A set $A$ of real numbers is said to be dense if every open interval contains a point of $A .$ For example, Problem 5 shows that the set of rational numbers and the set of irrational numbers are each dense. (a) Prove that if $f$ is continuous and $f(x)=0$ for all numbers $x$ in a dense set $A,$ then $f(x)=0$ for all $x.$ (b) Prove that if $f$ and $g$ are continuous and $f(x)=g(x)$ for all $x$ in a dense set $A,$ then $f(x)=g(x)$ for all $x.$ (c) If we assume instead that $f(x) \geq g(x)$ for all $x$ in $A,$ show that $f(x) \geq g(x)$ for all $x .$ Can $\geq$ be replaced by $>$ throughout?

Transcript

0:00 Hi.

00:01 In this problem, we're going to do a few problems related to dense sets and continuous functions.

00:08 So for a, we suppose that we have a continuous function f that's zero on a dense set a, and we want to prove that it's zero everywhere.

00:18 So what we do is contrapositive.

00:20 So we suppose there exists an x such that f of x is not equal to zero.

00:28 So f of x is going to equal some number.

00:33 So what we can do is we can separate f of x from zero.

00:38 So we can let you be an open set in r such that f of x is in you and zero is not in you.

00:55 So you imagine we have the number line and we have zero f of x and we can let you, be some open set like this.

01:06 It doesn't matter how close zero is to f of x.

01:10 Okay, but since f is continuous, f inverse of you is open.

01:22 But for all y in f inverse of you, f of y is going to be in u, of course, but since zero is not in you, that means f of y cannot equal 0.

01:41 Now this implies then that a, which we call the set x such that f of x equal to 0, is not dense.

01:55 So then by the contrapositive, if this is dense, then it has to be zero everywhere.

02:00 So we're done.

02:05 Now for b, well, one trick we can do is we can use a.

02:09 So we can let h of x be equal to f of x minus g of x.

02:19 Plus h of x is 0 on a dense set.

02:27 So by our work in a, h of x equals 0 everywhere.

02:36 But h of x is f of x minus g of x...

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