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Johnicholas comments on Mathematical simplicity bias and exponential functions - Less Wrong
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And if you use the ArthurB definition of "approximately" (which is an excellent definition for many purposes), then a piecewise constant function would do just as well.
Indeed.
But I may have gotten "scale" wrong here. If we scale the error at the same time as we scale the part we're looking at, then differentiability is necessary and sufficient. If we're concerned about approximating the function, on a smallish part, then continuous is what we're looking for.