It's called the “de la Vallée-Poussin derivative” or “Peano derivative” (both terms seem to have been used, sometimes with a difference between them; the latter is perhaps slightly more standard): see this answer and the references it contains for a proof of the fact that if the $k$-th Peano derivative exists and is continuous on an interval then it is, in fact, a $k$-th derivative.

It's called the “de la Vallée-Poussin derivative” or “Peano derivative” (both terms seem to have been used, sometimes with a difference between them; the latter is perhaps slightly more standard): see this answer and the references it contains for a proof of the fact that if the $k$-th Peano derivative exists and is continuous on an interval then it is, in fact, a $k$-th derivative.

Edit: The term “de la Vallée-Poussin derivative” seems to have been introduced in a paper by Marcinkiewicz and Zygmund, “On the Differentiability of Functions and Summability of Trigonometrical Series”, Fund. Math. 26 (1936) 1–43. The term “Peano derivative” occurs in Oliver's paper “The Exact Peano Derivative”, Trans. Amer. Math. Soc. 76 (1954) 444–456, alongside dlVP, but does not provide an explanation of where the author got this name.