Definition 1.4 of “A lambda calculus for real analysis” (Paul Taylor) says:
Definition 1.4 We say that $f:\Bbb R\to \Bbb R$ doesn't hover if, $$ \text{for any $e<t$,}\qquad \exists x.(e<x<t)\land (fx\ne 0)$$ so the open non-zero set $W_f\equiv \{x\mid fx\ne 0\}$ is dense.
(Page 7.)
I am perplexed by the behavior $t$ in this definition. It appears to be a free variable, but it does not appear free in the expression of the property that is being defined. Is the phrase “doesn't hover” a colloquialism for “doesn't hover (near $t$)”, or an abbreviation for “doesn't hover (for any given $t$)”, or something of that sort?
Later examples in the same source haven't helped me understand what taylor means by this.
Noah Schweber suggests that “For any $e<t$...” is short for “For any $e,t$ with $e<t$...”. Obvious once it's pointed out, but until it was, I was stumped.
Note that this is clearly equivalent to the following clause: the set $\{ x\mid fx≠0\}$ is dense, since every interval $(e,t)$ contains a point at which $f(x)$ is nonzero, as Dan Doel points out.
Thanks to both of you.