And as long as we are going for perfect rigor, the function you describe is undefined at x=0; you have to make it piecewise and say "if x=0, then this function is 0". (And I could say something about "you want natural curves, not this artificial pieced-together stuff.")

Also, I believe the Taylor series for that function is well-defined and accurate at all points other than x=0, so you can just pick one of those points. Are there smooth functions whose Taylor series are wrong everywhere? I doubt it. And one might argue that it's pretty problematic that, e.g., with the function "0 if x≤0, e^{-1/x^2} if x>0", a Taylor series at any positive point will be wrong for all negative numbers, and a Taylor series at any nonnegative point will be wrong for all positive numbers... well, I dunno, touching half of the entire range of a curve is still a lot (an infinite amount, in fact).

One could imagine a bunch of functions pieced together to make a smooth function (e.g. "e^{-1/x^2} * e^{-1/(x-1)^2} for 0<x<1, e^{-1/(x-1)^2} * e^{-1/(x-2)^2} for 1<x<2, ...") whose Taylor series are all accurate only over a small, finite domain. I guess that kinda answers my question. But at this point I would apply the "natural curves" objection. I wonder, are there smooth non-piecewise functions whose Taylor series are all wrong like that? (Absolute value is a piecewise function, so I'd probably reject anything with absolute value.)