A thought I had on a roadtrip not too long ago, which is not exactly true, but close enough:
If the white line on the side of a road is real analytic- then every twist and turn on the road should be discernable by investigating just an arbitrarily small neighborhood of a line. Thus, all 3,365 miles of the path that Route 20 takes from Kendall Square in Boston to Newport, Oregon could be deduced if one measured, precisely, a 1-foot section of the white line in Sioux City, Iowa.
What I am leaning on here is the definition of a real analytic function (and in light of my recent post on functions, I’ll specify that I am talking about real-valued functions on the real line): that a function has a convergent power series in a neighborhood of any point. One should also worry about the difference between an analytic function, and one which is merely smooth. A good example to remember is the function
One must take my word that this function is smooth (though it is easy enough to believe- note that the exponential function and all of its derivatives will be equal to 0 at 0, making a sort of smooth splice). However, note that the Taylor series at x= 0 is identically zero (as we just pointed out), so the Taylor series does not converge to the function for x>0. Hence f is a smooth, but non-analytic, function.
I mentioned that this statement was not exactly true. Mathematically, I meant it would be hard to verify that all 3,365 miles were real analytic, unless you could write down the formula for the road, or knew a priori that paint only dried in a smooth fashion. I prefer my brother’s objection when I told him the story though: exits. When I moved my story to the center line, he moved his to passing lanes. Ah well.