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## Smooth functionIn mathematics, asmooth function is one that is infinitely differentiable, i.e., has derivatives of all finite orders.For example, the exponential function is trivially smooth because the derivative of the exponential function is the exponential function itself. It is often useful to construct smooth functions that are zero outside a given interval, but not inside it. This is possible; on the other hand it is impossible that a power series can have that property. This shows that there is a large gap between smooth and analytic functions; so that Taylor's theorem cannot in general be applied to expand smooth functions.
To give an explicit construct of such functions, we can start with a function such as f(
Thinking in terms of complex analysis, a function like g(
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