Functions that change only for integers

Question

We all know there is floor function $[x]$ changes only for integers i.e.

$$[4]=[4.3]=[4.8]=4 ,[5]=5$$

I wonder if there are other functions that change only for integers? For instance we add $1$ for every integer to calculate floor function. So maybe we add $\ln x$ for every integer x to calculate "the function".

Note: I want a function that involves "elementary" functions like $log,sin,cos,...$

Answer 1

Take any function $\,f(x)\,$ that is not constant over the integers, then $\,f\left(\lfloor x \rfloor\right)\,$ will have jump discontinuities at integer points. Below is for example the graph of $\,\sin\left(\lfloor x \rfloor\right)\,$ .