Looking for an analytical function with f(n) = 1 with x=1 and f(n)=a otherwise.

Question

I am looking for a analytical function that applies to the constraints: \begin{equation}f(x) = \begin{cases}1 \text{ if } x=1 \\ \alpha \text{ otherwise }\end{cases}, \text{where } 0 \leq \alpha \leq 1 \text{ and } x\in\mathbb{N} \text{ with } x \geq 1\end{equation} I've been trying to figure this out with the help of WolframAlpha and all the Maths-Knowledge I have, but can't get there.

Links to already existing answers I couldn't find appreciated too.

Answer 1

How about $$ f(x)=\begin{cases}\alpha+(\alpha-1){\sin\pi x\over \pi (x-1)}&,\quad x\ne 1\\1&,\quad x=1\end{cases} $$?