To analyze the given table and determine the correct statements about the function it represents, we proceed as follows:

1. Domain of the Function:
- The table specifies the function's value for different intervals of $x$. These intervals are $(0, 5)$, $[5, 10)$, $[10, 15)$, and $[15, 20]$.
- Combining these intervals gives us the entire domain of the function, which is $(0, 5) \cup [5, 10) \cup [10, 15) \cup [15, 20] = (0, 20]$.
- Therefore, statement A, which says "The domain of the function is $(0, 20]$", is true.

2. Range of the Function:
- For each interval of $x$, the function assigns a specific value to $y$. These values are $2$, $3$, $4$, and $5$.
- The collection of these values forms the range of the function, which is $\{2\} \cup \{3\} \cup \{4\} \cup \{5\} = \{2, 3, 4, 5\}$.
- This means statement B, "The range of the function is $[2, 5]$", is incorrect because the range is not an interval but a set of discrete values. Hence, statement C, "The range of the function is $\{2, 3, 4, 5\}$", is correct.

3. Monotonicity of the Function:
- The function assigns different values to $y$ for different intervals of $x$, but it does not continuously increase or decrease across the entire domain. It jumps from one value to another as $x$ changes intervals.
- Therefore, the function does not exhibit monotonicity, making statement D, "The function is increasing", incorrect.

Given these analyses, the correct choices are A and C, encapsulated as $\boxed{\text{A and C}}$.