Definition of piecewise continuous/piecewise smooth

• AlonsoMcLaren
In summary, piecewise continuous refers to a function that is continuous on each individual piece of its domain, while piecewise smooth refers to a function that is differentiable on each individual piece of its domain. These types of functions are different from their respective counterparts, continuous and smooth functions, in that they are only continuous or differentiable on each piece of their domain rather than the entire domain. They have various real-world applications, including modeling temperature and stock prices, as well as in signal processing and control systems.

AlonsoMcLaren

In my textbook, piecewise continous/piecewise smooth is always defined on interval[a,b]. Can they be defined on open interval (a.b)?

Sure, I see no reason why not...

What is the definition of piecewise continuous?

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.

What is the definition of piecewise smooth?

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.

How is a piecewise continuous function different from a continuous function?

A continuous function is continuous on its entire domain, while a piecewise continuous function is only continuous on each individual piece of its domain.

How is a piecewise smooth function different from a smooth function?

A smooth function is differentiable on its entire domain, while a piecewise smooth function is only differentiable on each individual piece of its domain.

What are some real-world applications of piecewise continuous and piecewise smooth functions?

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.