mathematical function which rounds down k to n unless k > n+1, where n is integer

Question

So the floor function would round down k to n if k >= n+1, is there a commonly used name/symbol for a function where it rounds down is k > n+1? So for example,

f(4)=3

f(4.0001)=4

Answer 1

As requested in the comments:

This is one less than the ceiling function. Thus $f(x)=\lceil x \rceil -1$.