Is there a step function with an "in between value" that can be represented by a limit?

Question

I would be interested if there is a function $f(x)$ with the property that when a limit on the parameters of the function is be performed you get a step function with an in between value. What I mean by that is, assuming the function depends on a parameter a, you get $$ \lim_{a\rightarrow \infty} f(x) = \begin{cases} 0\quad x<x_0 \\ c \quad x=x_0 \\ 1 \quad x>x_0 \end{cases} $$ where $x_0$ could be $0$ for example and $0<c<1$.

Any help would be appreciated!

Answer 1

Assuming you want the convergence to be pointwise,

$$ f(a,x) = \begin{cases} \left(\frac12\sin(x)\right)^a\quad\text{if}\ x<x_0 \\ c-\frac{1}{a} \quad\text{if}\ x=x_0 \\ 1+\left(\frac12\sin(x)\right)^a \quad x>x_0 \end{cases} $$

If you want your function to be continuous for each value of $a$, then the question is more interesting. But you didn't mention this in your question.

Answer 2

With parameter $p\neq 0$ $$f(x)=\frac{1}{1+(\frac{1}{c}-1)e^{-p^2(x-x_0)}}$$ $f(x\to-\infty)\to 0$

$f(x_0)=c$

$f(x\to\infty)\to 1$

$p\to \infty \qquad \begin{cases} f(x<x_0)\to 0\\ f(x=x_0)=c\\ f(x>x_0)\to 1 \end{cases}$