Hensel's lemma for general continuous functions

In the present paper, we generalize the well-known Hensel's lifting lemma to any continuous function $f : \mathbb{Z}_p\rightarrow \mathbb{Z}_p$. This answers a question posed by Axelsson and Khrennikov (2016) who showed the validity of Hensel's lemma for $1$- and for $p^\alpha$-Lipschitz functions. For the statement and the proof, we introduce a suitable generalization of the original van der Put series. We use the concept of approximability of continuous functions to give numerical examples.