Graph of the Function $y = e^{-1/x^2}$

$y = e^{-1/x^{2}}$ is a function that becomes extraordinarily flat at the origin. Defining its value at $x = 0$ to be $0$ makes it differentiable any number of times, and yet it admits no Taylor expansion about that point. It is the standard example of the decisive difference between real and complex analysis.

Domain and range

The formula itself is defined for $x \neq 0$, but as $x \to 0$ we have $-\dfrac{1}{x^{2}} \to -\infty$ and hence $y \to 0$, so setting $f(0) = 0$ makes the function continuous on the whole real line, and that is the definition used here. As $|x| \to \infty$ the exponent approaches $0$ and $y \to 1$ without reaching it, so the range is $0 \leq y < 1$.

Symmetry and monotonicity

Since $x$ appears only squared, the function is even and symmetric about the $y$-axis. The derivative is $y' = \dfrac{2}{x^{3}}e^{-1/x^{2}}$, positive for $x > 0$ and negative for $x < 0$, so the origin is the minimum and the curve rises on both sides. The value at the bottom is $0$.

Concavity

The second derivative is $y'' = \dfrac{4 - 6x^{2}}{x^{6}}e^{-1/x^{2}}$, whose sign is that of $4 - 6x^{2}$, changing at $x = \pm\sqrt{\dfrac{2}{3}} \approx \pm 0.816$. Those two points are the inflection points, where the value is $e^{-3/2} \approx 0.223$; the curve is concave up between them and concave down outside.

How flat it is

It approaches $0$ faster than any power $x^{n}$, so near the origin the graph appears glued to the $x$-axis.

Every derivative vanishes at the origin

Differentiating repeatedly always produces something of the form $P\!\left( \dfrac{1}{x} \right)e^{-1/x^{2}}$, where $P$ is a polynomial. However large $\dfrac{1}{x}$ becomes, the decay of $e^{-1/x^{2}}$ outpaces it, so the product tends to $0$ as $x \to 0$. Hence $f'(0) = f''(0) = \cdots = 0$: derivatives of every order vanish at the origin.

No Taylor expansion

If every derivative is zero, the Maclaurin series about the origin is identically zero. That series converges for every $x$, but it agrees with $f(x)$ at the single point $x = 0$. Here is a function differentiable infinitely often whose Taylor series simply fails to represent it. Being infinitely differentiable is called $C^{\infty}$ and agreeing with one's own Taylor series is called analytic, and this function is $C^{\infty}$ without being analytic.

Contrast with complex analysis

Extending $e^{-1/z^{2}}$ to complex arguments makes the origin an essential singularity, near which the function takes almost every value. What looked like a flat point over the reals wears a completely different face once one steps into the complex plane.

Applications

Building on this function one can construct infinitely differentiable functions that are exactly zero outside an interval and positive inside it. The standard example takes $e^{-1/(1-x^{2})}$ for $|x| < 1$ and $0$ elsewhere, and is called a bump function. No polynomial or trigonometric function can vanish identically outside a bounded set in this way. Bump functions are the basis for partitions of unity in differential geometry and for the mollifiers used to smooth functions in partial differential equations, making them the standard tool for lifting local constructions to global ones.