[The nLab page on partitions of unity] 1 mentions the application of partitions of unity as a way to construct continuous maps to geometric realizations of simplicial spaces. However I often feel uncomfortable with the continuity of maps constructed in the realm of this example.

Let me discuss my uncertainty with the help of an explicit example.

Let $X$ be a space with an open cover $X=\bigcup_{i\in I} U_i$ and a partition of unity $\{f_i\colon X\rightarrow\mathbb{R}\}_{i\in I}$ subordinate to this cover and $\{a_i\}_{i\in I}$ a sequence of arbitrary real numbers.

Consider the poset $(\mathbb{R},\le)$ as a category internal to Spaces with the usual Euclidian topology. The nerve of this is a simplicial space $N_\bullet(\mathbb{R},\le)$. It has a geometric realization $||N_\bullet(\mathbb{R},\le)||=B(\mathbb{R},\le)$, whereby I mean the so called "fat" geometric realization, i.e. I only factor out the face maps and not the degeneracies.

Now define a map $g\colon X\rightarrow||N_\bullet(\mathbb{R},\le)||$ by mapping $x$ to the residue class of $((a_{i_0}\le...\le a_{i_k}),(f_{i_0}(x),...,f_{i_k}))\in N_k(\mathbb{R},\le)$ where ${i_0,...,i_k}$ are exactly those indices where $f_i(x)\neq0$.

This gives a well-defined map. Is this map always continuous? The choice of the non-zero $f_i(x)$'s feels somehow uncomfortable to me. The values of $g$ seem to "jump" somehow.

However, my impression that many of the constructions summarized by the mentioned example of the nlab go along this example, so probably my feeling is wrong and I would be happy seeing somehow thin out my fog.