This distinction is the difference between continuity and uniform continuity. Let

(i) is continuous on *I* iff

(∀*ε*>0)(∀*x*∈*I*)(∃*δ*>0)(∀*y*∈*I*)(|*x*−*y*|<*δ*⇒|(*x*)−(*y*)|<*ε*)

(ii) is uniformly continuous on *I* iff

(∀*ε*>0)(∃*δ*>0)(∀*x*∈*I*)(∀*y*∈*I*)(|*x*−*y*|<*δ*⇒|(*x*)−(*y*)|<*ε*)

The two definitions are different. In (i) *δ* depends on *x*; different *δ* may need to be chosen for different *x*. In (ii), one single *δ* has to work for all *x*.

Statement 2: For all a ∈ A there exists d ∈ D such that H(a, d).

This one I can't figure out. Does it mean that for every a there exists a unique d...or does it mean that d's can be shared by some a's...?

Since Statement 2 doesn't say "there exists a unique d", I would interpret this in the same manner as for Statement 1; namely, that there exists some element d for each a. The element d doesn't have to be unique (a different d for each a).

]]>Let's assume the following:

Two sets A and a D and a predicate H(a, d), where a ∈ A and d ∈ D.

Statement 1.

There exists d ∈ D for all a ∈ A such that H(a, d).

This one I can figure out. It means that there exists a single d for all a such that H(a, d).

Statement 2.

For all a ∈ A there exists d ∈ D such that H(a, d).

This one I can't figure out. Does it mean that for every a there exists a unique d, i.e a1 - d1, a2 - d2, a3 - d3 etc. or does it mean that d's can be shared by some a's, i.e. a1 - d1, a2 - d2, a3 - d1, a4 - d2 etc.

Any help is appreciated.

]]>