I was practicing function domains and function zeros

For example, this function:

$f(x) = x e^{\frac{1}{x}}$

It's domain is $\{x\in \mathbb{R} : x\neq0 \}$

It's function zero:

$f(x) = 0$

$x e^{\frac{1}{x}} = 0$

$x = 0$ ?

This is where my confusion comes in. Does this function have a zero?

I think it doesn't because $0$ is excluded from the domain... but i am unsure...

Thank you for your help :)

edit: I just realized how stupid of a question I had asked... there is a point in calculating function zero $f(x) = 0$, in my case the result of the function zero is $0$, but it's not in the domain so the function does not have a zero.

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    $\begingroup$ The function vanishes nowhere. It is negative for $x<0$ and positive for $x>0$. $\endgroup$ – egreg Oct 14 '18 at 21:44

Since $0$ doesn't belong to the domain of the function, it vanishes nowhere, because $e^{1/x}>0$ for every $x\ne0$.

In particular, $f(x)<0$ for $x<0$ and $f(x)>0$ for $x>0$.

One has $$ \lim_{x\to0^-}f(x)=0 $$ and $$ \lim_{x\to0^+}f(x)=\infty $$ so the function cannot be extended by continuity at $0$.

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