There's no shortage of functions $f:\mathbb{Z}\to\mathbb{R}$, or perhaps more generally $f:\mathbb{N}\to\mathbb{C}$, which then later get defined over some continuum like $g:\mathbb{R}\to\mathbb{R}$ or $g:\mathbb{C}\to\mathbb{C}$ such that $g(n)=f(n)$ for $n\in\mathbb{N}$.
Although there is imaginably no formal way to do this "nicely" for all such functions $f(n)$, I'd like to compile a general list of properties that $g$ should obey.
Some classic examples of functions, $f$, and their extension, $g$, include $$ \begin{align*} f(n)=\sum_{k=1}^n\frac{1}{k}&\iff g(x)=\psi(x+1)+\gamma,\\ f(n)=n!&\iff g(x)=\Gamma(x+1),\\ f(n)=F_n &\iff g(x)=\frac{\phi^x-(-\phi)^x}{\sqrt{5}}, \end{align*} $$ where $F_n$ is the $n^{\text{th}}$ Fibbonacci number, $\phi$ is the golden ratio, $\psi$ is the digamma function, and $\gamma$ is the Euler-Mascheroni constant.
There are, of course, infinitely many other choices for $g$, given any $f$, but there is often an intuitive way to restrict ourselves to one which fulfils certain additional properties. Here are some general observations of properties of $g$ I've noticed.
Some functional equation: For example, the fact that $(n+1)!=n!(n+1)$ is carried over to the Gamma function, and for sums like $f(n)=\sum_{k=1}^n s_n$, we often have that $g(x+1)=g(x)+s_{x+1}$.
Continuity or differentiability: Self explanatory, as probably the most important new properties allowed by extending the domain to a continuum.
Log-convexity: For the example of the Gamma function, it is the only log-convex function over $\mathbb{R}$ which interpolates the factorials and obeys the same functional equation.
Holomorphism: As with the Gamma function, again, it is effectively the only holomorphic function over on the positive complex half-plane which obeys the same functional equation and remains bounded for $1<\Re(x)<2$.
So my question(s) is/are
What other properties have you noticed fulfilled by various functions $g$?
When do we care about certain properties like continuity, differentiability, log-convexity or it being holomorphic?
Is there some notion of a meta-algorithm for appending desired properties until we are sure there exists only one such extension?