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In my textbook, piecewise continous/piecewise smooth is always defined on interval[a,b]. Can they be defined on open interval (a.b)?
Piecewise continuous refers to a function that is continuous on each individual piece of its domain, but may have discontinuities at the points where the pieces meet.
Piecewise smooth refers to a function that is differentiable on each individual piece of its domain, but may have points where the derivatives of the pieces do not match up at the points where the pieces meet.
A continuous function is continuous on its entire domain, while a piecewise continuous function is only continuous on each individual piece of its domain.
A smooth function is differentiable on its entire domain, while a piecewise smooth function is only differentiable on each individual piece of its domain.
These types of functions are often used in modeling real-world phenomena, such as temperature over time or the behavior of stock prices. They can also be used in signal processing and control systems.