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It is well-known that the usual 'epsilon-delta' definition of continuity is equivalent to the sequential definition (assuming countable choice). Less well-known is the sequential definition of uniform continuity:

a function $f:[0,1] \rightarrow \mathbb{R}$ is sequentially uniformly continuous if for any sequences $(w_{n})_{n\in \mathbb{N}}$, $(v_{n})_{m\in \mathbb{N}}$ in $[0,1]$ such that $\lim_{n\rightarrow \infty}|w_{n}- v_{n}|=0$, we have $\lim_{n\rightarrow \infty}|f(x_{n}-f(y_{n})|=0$.

Are there (nice) sequential definitions of related classes, like Lipschitz or similer concept? I could not immediately find any.

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    $\begingroup$ The Lipschitz condition $|f(x)-f(y)|\le L |x-y|$ is so much easier than (uniform) continuity -- why do you want a sequential characterization? $\endgroup$
    – Jochen Wengenroth
    Commented Apr 19 at 12:33
  • $\begingroup$ @JochenWengenroth In short: the sequential definitions (but not the epsilon-delta ones) have interesting logical properties. Not so short: reverse math is a research program where one identifies the minimal axioms needed to prove a given theorem of ordinary math. There is a certain range of logical systems (hyperarithmetical analysis) that is very hard to capture; sequential definitions greatly help to zoom in on hyperarithmetical analysis. $\endgroup$
    – Sam Sanders
    Commented Apr 19 at 14:32

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