I have often heard the following arguments made about the requirement for differentiable functions to be smooth:
- Fractal Functions are nowhere smooth
- Differentiable functions need to be smooth
- Therefore, differentiable functions need to be smooth
I tried to find a formal definition of smoothness that provides some sort of "checklist" - for instance, if "conditions a,b,c" are met then the function is smooth else not smooth. But when reading links such as this (https://en.wikipedia.org/wiki/Smoothness), I could not find any such "checklist" for definitions of smoothness and differentiability of functions.
As a particular example, consider the popular example of the Weierstrass Function - this function is said to be:
- Nowhere smooth
- A continuous Function
- A Fractal Function
- Nowhere differentiable
Yet the main argument (https://math.berkeley.edu/~brent/files/104_weierstrass.pdf ) I keep coming across that shows why the Weierestrass Function is not differentiable does not seem to directly rely on the absence of "smoothness" :
It seems like the non-differentiability of this function is established using Properties of Calculus, and not using smoothness? The above proof shows that the derivative of the Weierstrass Function is equal to "infinity" everywhere - therefore does not exist.
- But could a proof have been made that "directly demonstrates that the Weierstrass function is non-differentiable because it is not smooth"?
- Can we show that the Weierstrass function is not smooth?
Can someone please comment on the explicit relationship between smoothness of a function and differentiability? Is there some result/theorem that demonstrates the requirement of differentiable functions to be smooth? How do we mathematically quantify the "smoothness" of a function?
Thanks!