Monotonic

A real function or sequence is called monotonic if it either constantly increases or decreases. Thus, the sequence $1, 2, 4, 8, 16, \ldots$ of powers of 2 is monotonically increasing because each term is larger than the previous. The function $f(x) = x^2$ is monotonically decreasing on the interval $(-\infty, 0]$ and monotonically increasing on the interval $[0, \infty)$ . However, the function $f(x) = x^2$ is not monotonic over the entire real line because it sometimes increases and sometimes decreases.

More formally, a function $f$ is monotonically increasing (resp. decreasing) if $a \leq b \Longrightarrow f(a) \leq f(b)$ (resp. $f(a) \geq f(b)$ . The function is strictly monotonic if, in addition, $a \neq b \Longrightarrow f(a) \neq f(b)$ .

A differentiable function is monotonically increasing (resp. decreasing) if and only if its derivative is nonnegative (resp. nonpositive).

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Monotonic