Thank you very much for your nice answer, Greg Martin.
I am reading "Calculus on Manifolds" by Michael Spivak.
I cannot understand visually and clearly what the following theorem is saying.
Please give me a non-trivial concrete example $A \subset \mathbb{R}^2$ and $\mathcal{O}$ and $\Phi$ which satisfies the four properties.
I wanna plot several $\phi$'s by Wolfram Engine to understand visually.
Here is an example of a differentiable ($C^1$) partition of unity for $\Bbb R^n$; it should still allow you to get a good visual sense of the functions. Define $$ B(x) = \begin{cases} 1-2x^2, &\text{if } |x|\le\frac12, \\ 2(1-|x|)^2, &\text{ if} \frac12\le|x|\le1, \\ 0, &\text{if } |x|\ge1. \end{cases} $$ Then a partition of unity for $\Bbb R$ itself is $$ \{ \dots, B(x+2), B(x+1), B(x), B(x-1), B(x-2), \dots\} = \{ B(x+j)\colon j\in\Bbb Z\}. $$ A partition of unity for $\Bbb R^2$ is $$ \{ B(x+j)B(y+k)\colon j,k\in\Bbb Z\}. $$ And in general a partition of unity for $\Bbb R^n$ is $$ \bigg\{ \prod_{i=1}^n B(x_i+k_i)\colon k_1,\dots,k_n\in\Bbb Z\bigg\}. $$