On: The exponential function is a miracle

[obastani]

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_functio...

Nevertheless, it is actually true that all complex differentiable functions satisfy this property, which is miraculous.

[petters → obastani]

> 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 → petters]

The 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

[posterboy → petters]

fyi, that's the heavyside function

[GlenTheMachine → obastani]

Well, 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.