It is well-known that given two real functions $f,g$ infinitely differentiable and with bounded derivatives $|f^{(n)}(x)| \le M$ and $|g^{(n)}(x)| \le M$ for all $x\in [a,b]$ and all $n=0,1,2\ldots,$ such that $f^{(n)}(a) = g^{(n)}(a) $ for all $n=0,1,2\ldots$ , then from Taylor's theorem it follows that $f(x) = g(x)$ for all $x\in[a,b]$.
Thinking graphically, this means that we cannot smoothly branch-off from $f$ at any point, without derivatives becoming unbounded, which means that we cannot 'glue' perfectly well two different curves at a single point without the derivatives of one of them running to infinity.
I'm trying to find an algebraic or geometric explanation of this amazing fact, that the two functions must be equal everywhere. So my question is: Is there an explanation that does not involve their explicit expansion into power-series?
2026-05-05 10:20:05.1777976405