What Defines a Piecewise Smooth Function?

[Jehannum]

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'?

 

[mfb]

"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.

 

[Stephen Tashi]

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".