Sequential definition of uniform continuity

Question

The equivalence of the “$\varepsilon-\delta$” definition of continuity and the sequential definition is often used, but I’m wondering if there is possibly a sequential definition of uniform continuity? At least one which is even remotely applicable.

Answer 1

The relevant sequential criterion is that a function $f:X\to Y$ is uniformly continuous iff for any sequences sequences $(x_n)$ and $(y_n)$ in $X$, $d_X(x_n,y_n)\to 0$ implies that $d_Y(f(x_n),f(y_n))\to 0$.

Suppose $f$. Let $(x_n)$ and $(y_n)$ be sequences in $X$ such that $d_X(x_n,y_n)\to 0$ and let $\varepsilon>0$. Choose $\delta>0$ such that $d_X(x,y)<\delta$ implies $d_Y(f(x),f(y))<\varepsilon$ and $N$ such that $n\geqslant N$ implies $d_X(x_n,y_n)<\delta$. Then $d_Y(f(x_n),f(y_n))<\varepsilon$ for $n\geqslant N$, so that $d_Y(f(x),f(y))\to 0$.

Suppose $f$ is not uniformly continuous, then there exists $\gamma>0$ such that for every $n\in\mathbb N$, there exist $x_n,y_n\in X$ such that $d_X(x_n,y_n)<\frac1n$ but $d_Y(f(x_n),f(y_n))\geqslant \gamma$. It follows that $d_X(x_n,y_n)\to 0$ but $d_Y(f(x_n),f(y_n))\not\to 0$.