Non-trivial explicit example of a partition of unity Does exist a non-discrete paracompact example where is possible to give a partition of unity with the functions defined explicitly for a specific non trivial cover of the space?

Here is another simpler example that may help you visualize what partitions of unity are. Consider the open cover of $\mathbb{R}$ given by $U = (-\infty,2)$ and $V = (-2,\infty)$. A partition of unity associated to $\{U,V\}$ could be given by $\{ f_U, f_V \}$, where
$$ f_U, f_V \colon \mathbb{R} \to [0,1]$$
$$ f_U (x) = \left\{ \begin{array}{ll}
             1 & \text{ if } x \leq -1 \\
             \frac{1-x}{2} & \text{ if } -1 \leq x \leq 1 \\
             0 & \text{ if } x \geq 1 \end{array} \right. $$
$$ f_V (x) = \left\{ \begin{array}{ll}
             0 & \text{ if } x \leq -1 \\
             \frac{x-1}{2} & \text{ if } -1 \leq x \leq 1 \\
             1 & \text{ if } x \geq 1 \end{array} \right. $$
Then it is clear that $f_U(x)+f_V(x) = 1$ for all $x$. The support of $f_U$ is contained in $U$ and the support of $f_V$ in $V$. Of course the condition that any $x$ has an open neighbourhood where only a finite number of the functions are different from cero is automatic here because there are only two functions.
