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Function whose all derivatives vanish at a point
The function
y
(
x
≠
0
)
=
e
−
1
/
x
2
,
{\displaystyle y(x\neq 0)=e^{-1/x^{2}},}
y
(
0
)
=
0
{\displaystyle y(0)=0}
x
=
0
{\displaystyle x=0}
In mathematics , especially real analysis , a real function is flat at
x
0
{\displaystyle x_{0}}
derivatives at
x
0
{\displaystyle x_{0}}
0 .
A function that is flat at
x
0
{\displaystyle x_{0}}
analytic at
x
0
{\displaystyle x_{0}}
constant in a neighbourhood of
x
0
{\displaystyle x_{0}}
Taylor series ).
An example of a flat function at 0 is the function such that
f
(
0
)
=
0
{\displaystyle f(0)=0}
f
(
x
)
=
e
−
1
/
x
2
{\textstyle f(x)=e^{-1/x^{2}}}
x
≠
0.
{\displaystyle x\neq 0.}
The function need not be flat at just one point. Trivially, constant functions on
R
{\displaystyle \mathbb {R} }
f
(
x
)
=
0
{\displaystyle f(x)=0}
x
≤
0
{\displaystyle x\leq 0}
f
(
x
)
=
e
−
1
/
x
2
{\textstyle f(x)=e^{-1/x^{2}}}
x
>
0.
{\displaystyle x>0.}
The function defined by
f
(
x
)
=
{
e
−
1
/
x
2
if
x
≠
0
0
if
x
=
0
{\displaystyle f(x)={\begin{cases}e^{-1/x^{2}}&{\text{if }}x\neq 0\\0&{\text{if }}x=0\end{cases}}}
is flat at
x
=
0
{\displaystyle x=0}
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 .
Glaister, P. (December 1991), A Flat Function with Some Interesting Properties and an Application , The Mathematical Gazette, Vol. 75, No. 474, pp. 438–440, JSTOR 3618627 
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