Find values $f(0)=p$ to make $f(x)=e^{\frac{-1}{|x|}}$ continuous at $x=0$ Given a function $f(x)=e^{\frac{-1}{|x|}}$ when $x \neq 0$ and $f(x)=p$ if $x=0$. I need to find the values for $p$, such that the function becomes continuous at $x=0$. In order to be continuous at $x=0$, it needs to be defined at that point (which is clearly the case), also the limit $\lim_{x\rightarrow 0} f(x) $ must exist and $\lim_{x\rightarrow 0} f(x) = f(0)=p$ must hold.

So far as I see it, the limit does not depend on the value I take for $p$, so it must be unique, since $\lim_{x\rightarrow 0} f(x) $ can only take one value. However the question clearly states that I have to find valueS for $p$ (quoting: "Find the values of $p$ that make $f$ continuous at $0$.").

Am I wrong or is the question malformed?
The question is well formed, and you already found one part of the answer: there is only one possible value $p$ for which $f$ will be continuous.

I have encountered this type of question quite often, and usually, the use of the plural does not mean that there need to be several answers. Plural is used to denote that it is not clear how many answers there are (and there could also be no answer at all!)
The question is well-posed. In mathematical texts the wording with plural s in a question about some solution set usually does not imply any assumption about it's size. It has this answer:
$$
\lim_{x\to 0} e^{-1/|x|} = 0 = p \Rightarrow S = \{ 0 \}
$$