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In basic calculus the primary functions we learn about are polynomials, the exponential function, logarithms, and trigonometric functions. Then we learn about Maclaurin series and more generally Taylor series and we are led to believe that for any function f(x) and any point a in its domain you can find an interval around a on which the function is equal to its Taylor series computed at a.
For many of the elementary functions this is indeed the case. There are a number of fine details to be checked: the function needs to be defined and infinitely differentiable for all Taylor series coefficients to be defined.
One also needs to determine the interval of convergence of the Taylor series. For most examples that are given this interval of convergence is not equal to zero.
Let us call a function "smooth" if the derivatives of all orders are defined at every point in its domain. We will study some smooth functions that have points where the
Taylor series at these points differ from the function itself:
the Taylor series predicts nothing about the function values.
We will start with the particular example of the function flat(x)
flat(x) = 0 for x <= 0
= exp(-1/x) for x>0...