A 単調関数 is a type of mathematical function that consistently moves in one direction. This means that as the input values (often referred to as the independent variable) increase, the output values (the dependent variable) either consistently increase or consistently decrease, but never both. In formal terms, a function |f(x) - f(y)| is called 単調増加 if, for any two values x₁ and x₂ in its domain, if x₁ < x₂, then f(x₁) <= f(x₂). Conversely, a function is 単調減少 if, for any two values x₁ and x₂, if x₁ < x₂, then f(x₁) >= f(x₂).
単調関数は、さまざまな分野で重要です。 calculus, データ分析, and algorithm design, because they maintain a consistent relationship between input and output. For instance, in optimization problems, knowing that a function is monotonic can simplify finding minimum or maximum values. These functions are also useful in the context of 機械学習 and AI, particularly when evaluating the performance of models, as they can indicate whether an increase in input features leads to a predictable change in output predictions.
It’s important to note that a function can be strictly monotonic if it either only increases or only decreases without any flat segments. For example, the function f(x) = x is strictly increasing, while f(x) = -x is strictly decreasing. In contrast, a function like f(x) = x² is not monotonic because it decreases on the interval (-∞, 0) and increases on (0, +∞).