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tut comments on Mathematical simplicity bias and exponential functions - Less Wrong
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Under the usual mathematical meanings of "continuous", "function" and so on, this is strictly false. See: http://en.wikipedia.org/wiki/Weierstrass_function
It might be true under some radically intuitionist interpretation (a family of philosophies I have a lot of sympathy with). For example, I believe Brouwer argued that all "functions" from "reals" to "reals" are "continuous", though he was using his own interpretation of the terms inside of quotes. However, such an interpretation should probably be explained rather than assumed. ;)
Mathematically he should have said "any C1 function". But if you are measuring with a tolerance level that allows a step function to be called exponential, then we can probably say that any continuous function is analytic too.