If for any $\varepsilon$ exists $\delta$, does that mean that for every $\delta$ exists $\varepsilon$?

Question

For any $\varepsilon \gt 0$ exists $\delta \gt 0$, does that mean that for any $\delta \gt 0$ exists $\varepsilon \gt 0$?

If $\delta$ depends on $\varepsilon$ such as $\delta = \frac 1 \varepsilon$, than we could wirite $\varepsilon = \frac 1 \delta$, or not?

Answer 1

No.

Consider the relation "$\delta$ is the mother of $\varepsilon$". Then for every $\varepsilon$ there is a $\delta$, but not the reverse.