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How can I create "switches" [the term may be new, but I'll explain it] for piecewise defined functions ? Suppose a function:

$$ f(x)=\begin{cases}\alpha\;,x\in D_1\\\beta\;,x\in D_2\end{cases}\\\text{wherein domain of f is D and range R, also }D_1\cup D_2=D$$

So, after creating switches $S_1, S_2$:

$$f(x)=S_1(x)\alpha+S_2(x)\beta$$

So that

$$S_1(x)=0\;\forall x\in D_2 \;and\;S_1(x)=1\;\forall x\in D_1 \\S_2(x)=0\;\forall x\in D_1 \;and\;S_1(x)=1\;\forall x\in D_2 $$

Afer some thought I created one for a simple case:

$$\LARGE \tan^{-1}{\frac 1x}=\cot^{-1}x-\pi.\left(\frac{1-\frac{|x|}x}2\right)$$

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I think you're looking for indicator functions. Otherwise, here's my attempt for the special case where $D_1 = (-\infty, c)$ and $D_2 = (c, \infty)$ for some $c \in \mathbb R$: $$ f(x) = \frac{(\beta - \alpha)|x - c| + (\beta + \alpha)(x - c)}{2(x - c)} $$

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