The classic example $f(x) = 0$ for $x\leq 0$, $f(x) = \exp(-1/x^2)$ for $x>0$ shows that we can "glue" a straight line and a curve, retaining infinite smoothness at the connection point.
So, what makes this function structurally different from, say, $\sin(x)$, which, by Taylor series, is uniquely determined by its value and all derivatives at $0$.