Smallest $k$ such that any 2022 numbers can be represented as sum of $k$ essentially increasing functions.

Step 1 (Decomposition Concept): A function $f:\mathbb{R}\to \mathbb{R}$ is essentially increasing if $f(x)\le f(y)$ for all $x\le y$ except for a finite set of exceptions. We want to represent a sequence of 2022 real numbers as the sum of $k$ essentially increasing sequences. This is equivalent to representing a finite sequence as a sum of monotonic sequences.

Step 2 (Worst-Case Construction): Let the sequence of 2022 numbers be strictly decreasing, e.g., $x_1>x_2>\cdots >x_{2022}$. For any essentially increasing sequence, we can have at most one step down per component without violating the growth constraint. Thus, a strictly decreasing sequence of length $N$ requires at least $N$ independent step functions to be represented. For $N=2022$, this shows that $k\ge 2022$ is necessary.

Step 3 (Sufficiency of $k=2022$): Any sequence of length 2022 can be trivially decomposed into a sum of 2022 step functions, where each step function handles a single transition. Since each step function is essentially increasing, the minimum number of functions required is exactly 2022. $■$