Continuous refers to the property of a function where there are no abrupt changes or discontinuities in its graph. This means that the function can be drawn without lifting the pen from the paper.
Periodic means that the function repeats itself after a certain interval, called the period. This means that the function will have the same values at regular intervals along the x-axis, creating a repeating pattern.
A piecewise differentiable function is made up of smaller, simpler functions that are defined over specific intervals. This means that the function may have different rules or equations for different parts of the domain. A regular differentiable function, on the other hand, has a single rule or equation that applies to the entire domain.
One example is the sawtooth wave function, which is a continuous function that repeats itself in a "sawtooth" pattern. It is piecewise differentiable because it is made up of linear segments that are connected at specific points.
Continuous periodic piecewise differentiable functions are useful in science because they can model real-world phenomena that exhibit repeating patterns, such as waves, vibrations, and oscillations. They also allow for more precise and accurate calculations and predictions compared to simpler functions.