Let \(f\) be a function mapping \(\mathbb{R}\) to \(\mathbb{R}\). Give the definitions of each of the following concepts in words, without using logical symbols or the word “not”.
\(f\) is surjective
\(f\) is not injective
\(f\) is uniformly continuous
\(f\) is not uniformly continuous
Solution
Every real number is the value of \(f\) at some real number.
There are two distinct real numbers at which \(f\) takes the same value.
For every positive \(\varepsilon\) there is a positive \(\delta\) such that whenever two real numbers differ by at most \(\delta\), their images under \(f\) differ by at most \(\varepsilon\).
There is a positive \(\varepsilon\) such that for every positive \(\delta\) there are two real numbers differing by at most \(\delta\) whose images under \(f\) differ by more than \(\varepsilon\).