In real analysis, a smooth function is infinitely differentiable at each point in its domain, while a real analytic function is, at each point in its domain, the limit of a convergent power series in a neighbourhood of that point. All real analytic functions are smooth, but there exist smooth real functions that are not real analytic, as given below. The existence of smooth but non-analytic functions represents one of the main differences between differential geometry and analytic geometry. In terms of sheaf theory, this difference can be stated as follows: the sheaf of differentiable functions on a differentiable manifold is fine, in contrast with the analytic case. Smooth real functions with domain R n {\displaystyle \mathbb {R} ^{n}} and with support of compact closure (bump functions) are non-analytic at each boundary point of the closure of its support. One of the most important applications of smooth functions with support of compact closure is the construction of so-called mollifiers, which are important in theories of generalized functions, such as Laurent Schwartz's theory of distributions. The functions below are generally used to build up partitions of unity on differentiable manifolds.
An example function
Definition of the function
Consider the function
f ( x ) = { e − 1 x if x > 0 , 0 if x ≤ 0 , {\displaystyle f(x)={\begin{cases}e^{-{\frac {1}{x}}}&{\text{if }}x>0,\\0&{\text{if }}x\leq 0,\end{cases}}}
defined for every real number x.
The function is smooth The function f has continuous derivatives of all orders at every point x of the real line. The formula for these derivatives is
f ( n ) ( x ) = { p n ( x ) x 2 n f ( x ) if x > 0 , 0 if x ≤ 0 , {\displaystyle f^{(n)}(x)={\begin{cases}\displaystyle {\frac {p_{n}(x)}{x^{2n}}}\,f(x)&{\text{if }}x>0,\\0&{\text{if }}x\leq 0,\end{cases}}}
where pn(x) is a polynomial of degree n − 1 given recursively by p1(x) = 1 and
p n + 1 ( x ) = x 2 p n ′ ( x ) − ( 2 n x − 1 ) p n ( x ) {\displaystyle p_{n+1}(x)=x^{2}p_{n}'(x)-(2nx-1)p_{n}(x)}
for any positive integer n. From this formula, it is not completely clear that the derivatives are continuous at 0; this follows from the one-sided limit
lim x ↘ 0 e − 1 x x m = 0 {\displaystyle \lim _{x\searrow 0}{\frac {e^{-{\frac {1}{x}}}}{x^{m}}}=0}
for any nonnegative integer m.
The function is not analytic As seen earlier, the function f is smooth, and all its derivatives at the origin are 0. Therefore, the Taylor series of f at the origin converges everywhere to the zero function,
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