Specific family of functions?

Question

Let $\Phi$ be the class of all fonctions $\phi$ (let's say from $\mathbb{R}^+$ to $\mathbb{R}^+$) such that:

If there exist an increasing sequences (not necessarily strictly) $\{t_n\}\subset \mathbb{R}^+$ that converges, then the sequence $\{\phi(t_n)\}$ converges also.

• Is there any specific name for this class of functions?

• It will be interesting to see some interesting examples/counterexamples.

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See similar discussion here and here.

Answer 1

$\phi \in \Phi$ if and only if

1. $f$ has left limits at all points, i.e. $\lim_{s<t, s \to t} f(s) $ exists and

2. $\lim_{t\to \infty} f(t)$ exists.

[The limit in 1) need not be $f(t)$ so $f$ need not be left continuous].

All monotone bounded functions have these properties.