I have since learned that there are functions which do have all partial derivatives at a point but are not smooth. Wikipedia’s example is f(x,y)=y3x2+y2 with f(0,0)=0. And in this case, there is still a continuous function ϕ:S2→R that maps each point to the value of the directional derivative, but it’s ϕ(x,y)=y3, so different from the regular case.
So you can probably have all kinds of relationships between direction and {value of derivative in that direction}, but the class of smooth functions have a fixed relationship. It still feels surprising that ‘most’ functions we work with just happen to be smooth.
That’s probably right.
I have since learned that there are functions which do have all partial derivatives at a point but are not smooth. Wikipedia’s example is f(x,y)=y3x2+y2 with f(0,0)=0. And in this case, there is still a continuous function ϕ:S2→R that maps each point to the value of the directional derivative, but it’s ϕ(x,y)=y3, so different from the regular case.
So you can probably have all kinds of relationships between direction and {value of derivative in that direction}, but the class of smooth functions have a fixed relationship. It still feels surprising that ‘most’ functions we work with just happen to be smooth.