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AxiomOfChoice
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Suppose I have an infinitely differentiable function F that is nonzero exactly on a set [-b,b]. Can I say that [itex]|F(x)| \leq C(x+b)^k[/itex] for some integer [itex]k > 2[/itex]? If so, why?
lugita15 said:Are you saying that F is infinitely differentiable on the open interval (-b,b) but not necessarily continuous at the two endpoints of the closed interval [-b,b]?
I feel a little bad about this, but I don't see why this is. Could you explain, please? Is it just because, since [itex]f(x) = 0[/itex] for [itex]x \leq -b[/itex], we must have [itex]f'(x) = 0[/itex] for [itex]x < -b[/itex], so by continuity of the derivative, [itex]f'(-b)[/itex] can't possibly be anything other than zero? (And then the same argument is repeated for the higher order derivatives.)jgens said:If [itex]f[/itex] is supported in [itex][-b,b][/itex], then certainly [itex]f^{(k)}(-b) = 0[/itex] for all [itex]k \in \mathbb{N}[/itex].
An infinitely differentiable function is a mathematical function that has derivatives of all orders at every point in its domain. This means that the function can be differentiated an infinite number of times without approaching a non-differentiable point.
A differentiable function is one that has a derivative at every point in its domain, while an infinitely differentiable function has derivatives of all orders at every point in its domain. In other words, an infinitely differentiable function is a more specialized type of differentiable function.
A function is considered to be infinitely differentiable if it has derivatives of all orders at every point in its domain. To determine this, you can use the definition of differentiability and take the derivative of the function multiple times. If the resulting derivatives exist and are continuous, then the function is infinitely differentiable.
Infinitely differentiable functions are important in mathematics because they provide a more detailed description of a function's behavior. They are used in many areas of mathematics, such as calculus, differential equations, and complex analysis, and are essential for solving many mathematical problems.
No, not all functions can be infinitely differentiable. For a function to be infinitely differentiable, it must have derivatives of all orders at every point in its domain. Some functions, such as the absolute value function, have non-differentiable points and are not infinitely differentiable.