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Ratzinger
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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?
For a function to be infinitely differentiable, it means that it can be differentiated an infinite number of times. This means that the function has derivatives of all orders and the derivatives are continuous. In other words, the function can be smoothly curved without any sharp corners or breaks.
Infinitely differentiable functions are important in many areas of mathematics and science because they allow for more precise and accurate modeling of real-world phenomena. They also have various applications in engineering, physics, and economics.
No, not all functions are infinitely differentiable. For a function to be infinitely differentiable, it must have a smooth and continuous graph without any sharp corners or breaks. Functions that have discontinuities or sharp corners, such as the absolute value function, are not infinitely differentiable.
A function that is continuously differentiable has derivatives of all orders, but those derivatives may not be continuous. On the other hand, a function that is infinitely differentiable has derivatives of all orders that are also continuous. In other words, a function that is infinitely differentiable is also continuously differentiable, but the reverse is not always true.
To prove that a function is infinitely differentiable, you must show that it has derivatives of all orders and that those derivatives are continuous. This can be done using mathematical techniques such as the limit definition of a derivative, the product rule, and the chain rule.