I am looking for a function $f(n,k)$ in which for $0\leq k\leq1$ the function image contains all the values of the interval $[1,nH_n)$, where $H_n = \sum_{k=1}^n \frac{1}{k}$ is the $n^{th}$ harmonic number. On the one hand, for the segment of the interval between 0 and $n$ the function $g(n,k)=n^k$ can be used, and $h(n,k)=n\cdot(H_{n})^k$ for the remaining values.
As seen, $g$ and $h$ span $[0,n]\cup [n,nH_n]$, however, as I want a single function to span such union, I tried with $f(n,k)=n(H_n-H_{n-n^k})$, which seems to perform well. However, 2 issues appear by using this form:
Since $H_n$ is asymptotically equivalent to $\log(n)$, can the harmonic terms be substituted by logarithms without affecting the continuity of $f(n,k)$ in the domain interval $k\in[0,1]$?
$H_n$ seems to be defined only for integer values, so, is there any way to prove that $f(n,k)$ is continuous in $k\in[0,1]$? Or, does the reciprocal form with logarithms suffice to span the $[1,nH_n)$ image values?
Also, when $n\to\infty$, $f(n,k)$ tends to 1 for all $k$ except the supremum value $k=1$. So, does the function still span the range $[1,nH_n)$ in such situation?