Determine intervals where a function is increasing or decreasing

A function is increasing on an interval when its output values rise as the input increases, and decreasing when its output values fall; these intervals can be identified from a graph, table, formula, or the sign of the first derivative. The analysis partitions the domain at critical numbers, discontinuities, and endpoints, distinguishing intervals of monotonic behavior from isolated turning points and recognizing that a zero derivative does not necessarily indicate a maximum or minimum. This treatment excludes abstract monotonicity on general domains and pathological functions.