What Defines a Piecewise Smooth Function?

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SUMMARY

A piecewise smooth function is defined as a function that is continuous but lacks smoothness due to the absence of derivatives of all orders. An example provided is f(x) = max(x, x²), which is continuous between (0,0) and (1,1) but not differentiable at x=1. The discussion highlights that while some functions are continuous everywhere, they may not be differentiable anywhere, such as the Weierstrass function. The term "piecewise smooth" accurately describes functions that exhibit this behavior.

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For example, if we define f(x) as "the greater of x and x2" it will give a straight line graph between (0,0) and (1,1) then turn into a curve. This function is continuous but not 'smooth'.

Is there any special name for this kind of function?

Are there any interesting considerations about such functions - or is it just a case of 'split them up into parts when necessary'?
 
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"Continuous but not differentiable"?
There are functions continuous everywhere, but not differentiable anywhere (e. g. the Weierstrass function).
Smooth is a mathematical term, and requires derivatives of all orders to exist.
 
Jehannum said:
Is there any special name for this kind of function?

The adjective "piecewise" could be used. (https://en.wikipedia.org/wiki/Piecewise ). You could say the function is "piecewise smooth".
 
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