Is there any name for this family of functions?

Question

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

For each sequence $\{t_n\}\subset \mathbb{R}^+$ that converges, 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.

Answer 1

There is a tiny gap between the stated condition and the standard definition of a continuous function, but that gap can be filled as follows: if $\{ t_n\}$ converges to $a$, consider the sequence $\{s_n\}$ with terms $t_1, a, t_2, a, t_3, a, \ldots $. Then $\{ s_n\}$ converges (to $a$) and the convergence of $\{ f(s_n) \}$ implies that $\{f(t_n)\}$ converges to $f(a)$.