Function collection with specific properties

Question

I'm looking for at least one answer to each number. If I know some function that holds I will put the answer, but, if there others, I would like to know too.

If is there's a similar function with the property described, I would like to know too.

These functions could be very useful to join formulas, describe completely some kind of properties and so many applications. Several times I need something similar to this, so I decided search and collect.

PS.: is prefereable to me functions defined in the complex domain or $f_{x}:\mathbb{C}\implies\mathbb{C}$, where x is the function number. But, probably will be interesting to the functions 1 to 6, answers defined on reals ($f_{x}:\mathbb{R}\implies\mathbb{R}$).

Is there a function that:

Is import to say, below we have some function definitions, but this isn't the kind of definition what I'm looking for. I'm searching for definitions who didn't use comparision to find its value. For example, $f_1(k)$ is equal to 1, as defined, but this value was found using comparision. But using the Kronecker Delta, we find this, without comparision. And this is the strong power of the Kronecer Delta. The full power of these functions can be viewed when applicating $f_6$ to some problems, for example.

1)$f_1(x)=\begin{cases} 0 & x\ne k \\ 1 & x= k \end{cases}$

2)$f_2(x)=\begin{cases} 0 & x< k \\ 1 & x\geq k \end{cases}$

3)$f_3(x)=\begin{cases} 0 & x< k \\ 1 & x> k \end{cases}$

4)$f_4(x)=\begin{cases} 0 & x< k \\ i & x= k \\ 1 & x> k \end{cases}$

5)$f_5(x)=\begin{cases} 0 & x\in(\mathbb{Q}-\mathbb{Z}) \\ 1 & x\in \mathbb{Z} \end{cases}$

6)$f_6(x)=\begin{cases} 0 & x\in \mathbb{Q}\\ 1 & x \in\mathbb{I} \quad(\mathbb{I}=\mathbb{R}-\mathbb{Q}) \end{cases}$

7)$f_7(x)=\begin{cases} 0 & x \in \mathbb{R}\\ 1 & x \in (\mathbb{C}-\mathbb{R}) \end{cases}$

8)$f_8(x)=\begin{cases} 0 & x \in \mathbb{R}\\ i & x=a+bi \quad (a,b\in \mathbb{R}-\{0\})\\ 1 & x=ai \quad (a\in \mathbb{R}-\{0\}) \end{cases}$

I'm not looking for pieceswise answers like $f(x)=\begin{cases} 0 & x< 0 \\ 1 & x\geq 0 \end{cases}$ that we need comput using comparision, but maybe special functions like Headvise step function($H(x) = \int_{-\infty}^x { \delta(t)} \, \mathrm{d}t$).

I will write the answers given or collected in the next section. Appreciate the help, thanks.

Bone collector of answers

1)

Kronecker Delta

$\delta_{x,k} = \frac1{2\pi i} \oint_{|z|=1} z^{x-k-1} dz=\frac1{2\pi} \int_0^{2\pi} e^{i(x-k)\varphi} d\varphi$

2)

Maybe a interpretation of Heaviside Step Function, not sure:

$H(x-k) = \int_{-\infty}^{x+k} { \delta(t)} \, \mathrm{d}t$

3)

4)

5)

6)

7)

8)

Answer 1

$$ f_2(x) = \frac{x-k}{\sqrt{(x-k)^2 + \epsilon^2}}+1 $$

This will get sharper as $\epsilon \to 0$.

Eg. Let $k = 3$, $\epsilon = 0$.