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That's not actually true. There exist functions of a single real variable that have derivatives at the origin that are all zero, yet are nonzero. For an example
by obastani 7y ago
That's not actually true. There exist functions of a single real variable that have derivatives at the origin that are all zero, yet are nonzero. For an example, see the following:
https://en.wikipedia.org/wiki/Taylor_series#Analytic_functions https://en.wikipedia.org/wiki/Taylor_series#Analytic_functio...
Nevertheless, it is actually true that all complex differentiable functions satisfy this property, which is miraculous.
- GlenTheMachine 7y agoWell, sure. Not all functions are described by their Taylor expansions. But this very large class is. I agree, if the word “miraculous” applies at all in mathematics, then this surely is one.
- petters 7y ago> There exist functions of a single real variable that have derivatives at the origin that are all zero, yet are nonzero The function that is equal to 0 for x<1 and equal to 1 otherwise also satisfies this.
- messe 7y agoThe more interesting statement is: > There exist smooth functions of a single real variable that have derivatives at the origin that are all zero, yet are nonzero https://en.wikipedia.org/wiki/Non-analytic_smooth_function https://en.wikipedia.org/wiki/Non-analytic_smooth_function
- posterboy 7y agofyi, that's the heavyside function