A monotonic function is a function that moves consistently in one direction—it either never decreases or never increases as the input increases. Monotonic functions can be either curved or linear, as long as the direction of change doesn’t reverse. Unlike linear functions, the change between values doesn’t have to be constant, which allows monotonic functions to bend.
There are two main types of monotonic functions:
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Monotonically increasing: The function’s output either stays the same or increases as the input increases.
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Monotonically decreasing: The function’s output either stays the same or decreases as the input increases.
A function is called strictly monotonic if it is always increasing or always decreasing, without any flat (constant) segments.
| Positive Monotonic: tends to increase but not necessarily in a linear fashion. |  |
| Negative Monotonic: Tends to decrease but not necessarily in a linear fashion. |  |
| Non-Monotonic: No overall tendency to either increase or decrease |  |
Monotonic functions are important in mathematics and statistics because they preserve order. If one input is larger than another, its output will also be larger (or smaller, in a decreasing function). This property is useful in ranking, comparing values, and transforming data in ways that do not alter the relationships between observations. Spearman’s rank order correlation can assess the strength of a nonlinear monotonic relationship.
For example, the natural logarithm function is monotonically increasing: as the input value increases, the logarithm also increases.
