g_edgar said:the derivative has the Darboux property... http://planetmath.org/encyclopedia/DarbouxsTheorem.html"
A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is defined as the slope of the tangent line to the function at that point.
Piecewise continuity of a derivative means that the derivative exists and is continuous on each interval of the function's domain. This means that the derivative is defined and has a finite value at every point within each interval.
A derivative has to be piecewise continuous because it is a fundamental requirement for the derivative to exist at a point. If a function is not continuous, its derivative will not exist at that point.
If a derivative is not piecewise continuous, it means that the function is not continuous at some point in its domain. This can lead to a variety of issues and inaccuracies in calculations that involve the derivative, making it difficult to accurately analyze the behavior of the function.
To determine if a derivative is piecewise continuous, you can check if the function is continuous on each interval of its domain. This can be done by evaluating the limit of the derivative as it approaches each point within the interval. If the limit exists and is finite, the derivative is piecewise continuous.