Is there a reason why this function does not exist/can't be found?

Question

I'm looking at a function $f\colon \mathbb N \rightarrow \mathbb R$, defined such that $(\Delta f)(x) = 1/x$. However, I know such a function does not exist or has not been found yet. I'm interested in why we can't find it. Is it because we lack the sufficient tools? Has it been proven not to exist?

Note: Edited again to restrict the domain

Answer 1

Define $f(x):=\int_0^1\dfrac{1-t^{x-1}}{1-t}dt$ so $\Delta f(x):=f(x+1)-f(x)=\int_0^1 t^{x-1} dt=\dfrac{1}{x}$.

Answer 2

By definition, $\Delta f=f(n+1)-f(n)$, so you are looking for $$f(n+1)-f(n)=\frac{1}{n}.$$ So it looks like $$f(n)=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n-1}=H_{n-1}$$ should work.

Proof $$f(n+1)-f(n)=1+\frac{1}{2}+\cdots+\frac{1}{n}-\left(1+\frac{1}{2}+\cdots+\frac{1}{n-1}\right)=\frac{1}{n}.$$