smooth but not analytic

It is tempting to believe that if a function is infinitely differentiable, its Taylor series must rebuild it. That belief is false, and the counterexample is one of the most instructive in analysis. A function can be perfectly smooth — possessing derivatives of every order — and yet its Taylor series can converge to the wrong thing, missing the function entirely.

The canonical example is f(x) = e^{-1/x^2} for x not equal to 0, with f(0) = 0. This function is infinitely differentiable everywhere, and every single derivative at 0 turns out to equal 0: f(0) = 0, f'(0) = 0, f''(0) = 0, and so on for all orders. Therefore its Taylor series at 0 is 0 + 0x + 0x^2 + ... , the zero series, which sums to the constant 0 — but f(x) is positive for every x not equal to 0.

The lesson is that smoothness (membership in C-infinity) is strictly weaker than real-analyticity. The Taylor series of a smooth function always exists, but equality f(x) = (its Taylor series) is a genuine extra condition, controlled by whether the Taylor remainder tends to zero. Far from being a mere curiosity, such functions are indispensable: they build bump functions and partitions of unity, smooth tools that analytic functions (being rigid) cannot provide.