There is—for example—the piecewise function where $f(x) = e^{(-1/( x^2 ))}$ if $x \neq 0$ and $f(x) = 0$ if $x = 0$, where the Taylor Series (centered at $x = 0$) becomes $0 + 0x + 0x^2 + 0x^n +…$, not converging to $f(x)$. But, are there any, even if complicated, examples of functions that are not piecewise, whilst still having a converging Taylor Series.
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2what do you mean by 'non-piecewise'. Whatever you think this phrase means, I'm 99% sure it has no formal meaning. You will not be able to generate a non-analytic function simply by adding, subtracting, multiplying, dividing, composing, differentiating, integrating the good old functions we all know and love (polynomials, trigonometric functions, exponential, logarithm, their inverses), because these are all analytic functions and remain so after these operations (within the radius of convergence, things behave as nice as we expect). – peek-a-boo Mar 29 '23 at 05:47