 In mathematical prose, one can say things such as ‘for all real numbers \(a\) and for all positive real numbers \(\varepsilon\), there exists a positive real number \(\delta\) depending on \(\varepsilon\) but not on \(a\), such that…’ What is important here is quantifier dependence. The existential quantifier ‘there exists \(\delta\)’ is said to depend on the universal quantifier ‘for all \(\varepsilon\)’ but not on the universal quantifier ‘for all \(a\).’ It was an essential part of Karl Weierstrass’s (1815–1897) work in the foundations of analysis that he defined the notions of limit, continuity, and derivative in terms of quantifier  dependence.[1]  For a concrete example, a function \(f : D \rightarrow \mathbf{R}\) is continuous, if for all \(a\) in the set \(D\) and for all \(\varepsilon \gt 0\) there exists \(\delta \gt 0\) such that for all \(x\) in \(D\), if \(|x - a| \lt \delta\), then \(|f(x) - f(a)| \lt \varepsilon\). The definition of uniform continuity is obtained from that of continuity by specifying that the quantifier ‘there exists \(\delta\)’ depends only on the quantifier ‘for all \(\varepsilon\),’ not on the quantifier ‘for all  \(a\).’[2]