What is continuous piecewise functions?

Continuous piecewise functions are functions that are defined by multiple equations over different intervals. Each equation is defined over a specific interval or set of intervals, and these intervals are commonly known as "pieces". At each point where two pieces meet, the function must be continuous, meaning that the values on either side of the junction must be equal.

Piecewise functions are used to define functions that have different rules or formulas for specific ranges or domains of the input variable. For example, a piecewise function may be used to describe the temperature of water plotted against time. The temperature may be constant at 0 degrees Celsius until the water reaches boiling point. At that point, the function may switch to a new equation that describes the temperature change as the water changes state from liquid to steam.

These functions are often used in mathematical models and in practical applications such as finance, physics and engineering. They are also used in programming languages to define complex functions with multiple branches and to control the flow of programs.