What is the Importance of Infinitely Differentiable Functions in Mathematics?

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Infinitely differentiable functions, also known as smooth functions, can have all higher derivatives equal to zero at a point, such as in the case of certain functions defined piecewise. An example provided is f(x), which is zero at x = 0 and follows a different rule for x ≠ 0. The key aspect is that the function must be defined and differentiable at all orders, regardless of the values of the derivatives. This property is significant in various areas of mathematics, including analysis and differential equations. Understanding infinitely differentiable functions is essential for exploring concepts like Taylor series and smooth manifolds.
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infinitely differentiable doesn't care if all the higher derivatives are zeroes (like for polynomials), it only has to be defined...correct?
 
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Correct...
 
It's okay for them to all be zero as well!

such as for

<br /> f(x) = \left\{<br /> \begin{array}{ll}<br /> 0 &amp; x = 0 \\<br /> e^{-1/x^2} \quad &amp; x \neq 0 <br /> \end{array}<br />

at x = 0.
 

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