The word monotonically in monotonically increasing is in fact redundant. It says nothing more than increasing. Mathematicians could cut out this word by adopting increasing function in the place of monotonically increasing function without any loss of clarity.
If the function goes horizontal sometimes, then it is nondecreasing. If always, then it's constant (at least if continuous).
What I learned was that “increasing” can imply a continuous series of increments, whereas “monotonically” qualifies that to unambiguously restrict the term to a specific meaning, not the casual one.
Much jargon works this narrowing way, to demarcate that the concept should not carry the everyday cluster of associations and meanings, but rather a domain-specific meaning.
The value in the stock market is an increasing function, but neither nondecreasing, nor constant.
And most of all, not monotonically increasing.
The monotonically is important because it says that at every zoom level the function is increasing, while it being increasing just says it about the function shape in general.
If the function goes horizontal sometimes, then it is nondecreasing. If always, then it's constant (at least if continuous).