I while back, my calculus teacher said something that I find very bothersome. I didn't have time to clarify, but he said:
If a function is discontinuous, automatically, it's not differentiable.
I find this bothersome because I can think of many discontinuous piecewise functions like this:
f(x)={x2,x≤3x2+3,x>3
Where f′(x) would have two parts of the same function, and give:
f′(x)={2x,x≤32x,x>3=2x
So I'm wondering, what exactly is wrong with this? Is there something I'm missing about what it means to be ""continuous""? Or maybe, are there special rules for how to deal with the derivatives of piecewise functions, that I don't know about.
