A piecewise function is continuous on a given interval in its domain if the following conditions are met: its constituent functions are continuous on the corresponding intervals (subdomains), there is no discontinuity at each endpoint of the subdomains within that interval.

Does continuous imply piecewise continuous?

A piecewise continuous function doesn't have to be continuous at finitely many points in a finite interval, so long as you can split the function into subintervals such that each interval is continuous. The function itself is not continuous, but each little segment is in itself continuous.

Is a continuous function piecewise smooth?

If it is continuous, it is piecewise continuous (in one big piece). If it is piecewise smooth, then it needn't be piecewise continuous. For example, f(x)= |x| is "continuous and piecewise differentiable": it is continuous for all x and differentiable every where except at x= 0 so differentiable on the "pieces" and .