# Is there point in calculating a function zero if zero isnt in it's domain?

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.

• The function vanishes nowhere. It is negative for $x<0$ and positive for $x>0$. – 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 $$0$$.