A research on limit, continuum, and differentiability of piecewise function at piecewise point

Through diseussing of the properties and state of the piecewise function at the piecewise point, we figure out the usage of the piecewise function on helping student to understand some basic concerns in higher mathematics. We propose a method to judge if the derivative at the piecewise point exists by using the limit of derivative function at the piecewise point under some conditions.