A set $A$ of real numbers is said to be dense if every open intervalcontains a point of $A$.
My main confusion is with problem (c), I will post my proofs for (a) and (b) for background.
(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 $x$
Lemma: If $f(x_o)>0$ then there is some $\delta>0$ such that for every $x$ satisfying $x_o-\delta<x<x_0+\delta$, $f(x)>0$.
Suppose not, so $f(x_o)>0$ or $f(x_o)<0$ for some $x_o$ not in A. For simplicity consider $f(x_o)>0$ since the contrary is similar. By the lemma, there is some $\delta>0$ such that for every $x$ satisfying $x_o-\delta<x<x_0+\delta$, $f(x)>0$. But since A is dense there must be $x$ in $x_o-\delta<x<x_0+\delta$ that is also in $A$ such that $f(x)=0$. This is a contradiction, thus $f(x)=0$ for all $x$.
(b) Prove that if $f$ and $g$ are continuous and $f(x)=g(x)$ for $x$in a dense set $A$, then $f(x)=g(x)$ for all $x$.
This follows directly from (a) if you consider a function $h$ such that $h=f-g$. The next proof confuses me.
(c) If we assume instead that $f(x)\ge g(x)$ for all $x$ in $A$, showthat $f(x)\ge g(x)$ for all $x$. Can $\ge$ be replaced with $>$throughout?
The case where $f(x)=g(x)$ follows from (b). Now, consider $f(x)>g(x)$. We claim that this is true. Suppose not, so $g(x_0)>f(x_o)$ for some $x_o$. Define a function $q=g-f$ This implies $q(x_0)>0$. By the lemma, there is some $\delta>0$ such that for every $x$ satisfying $x_o-\delta<x<x_0+\delta$, $q(x)>0$. But since A is dense there must be an $x$ in $x_o-\delta<x<x_0+\delta$ that is also in $A$ such that $q(x)<0$. A contradiction, thus $f(x)\ge g(x)$ for all $x$. Q.E.D
My proofs seem correct but I must be missing something, because Spivak says
It is not possible to replace $\ge$ by $>$ throughout. For example, if$f(x)=|x|$, then $f(x)>0$ for all $x$ in the dense set $\{ x:x\neq0 \}$,but it is not true that $f(x)>0$ for all $x$.
I understand the counterexample, but this contradicts my proof to (c). I can't figure out how to re-adjust my proof to account for this. Any advice would be appreciated.Thanks!
