In mathematics, especially real analysis, a real function is flat at x_0 if all its derivatives at x_0 exist and equal 0 . A function that is flat at x_0 is not analytic at x_0 unless it is constant in a neighbourhood of x_0 (since an analytic function must equals the sum of its Taylor series). An example of a flat function at 0 is the function such that f(0)=0 and for x\neq 0. The function need not be flat at just one point. Trivially, constant functions on \mathbb{R} are flat everywhere. But there are also other, less trivial, examples; for example, the function such that f(x)=0 for x\leq 0 and for x> 0.

Example

The function defined by is flat at x = 0. Thus, this is an example of a non-analytic smooth function. The pathological nature of this example is partially illuminated by the fact that its extension to the complex numbers is, in fact, not differentiable.