There's a reason I've said "looks like" a smooth curve (I'm hesitant to say this but maybe one could define this as meaning it is a smooth curve using definitions of "geometric continuity")
I'm thinking of the following (limiting it to 2-Dimensions):
- I draw some curve on a paper, it "looks" continuous (has no gaps) and it "looks" smooth too ("looks" like it has continuous first-order derivatives)
- Let's say the title is not proven to be true, could it be that there exist curves, such as the one I drew, which "seem", visually, to be smooth but can't technically be as such (since there is no function to derive and assuming the definition of smoothness is continuous first-order derivatives)