Can a function with just one point in its domain be continuous?

For example if my function is $f:\{1\}\longrightarrow \mathbb{R}$ such that $f(1)=1$.

I have the next context:

1) According to the definition given in Spivak's book and also in wikipedia, since $\lim_{x\to1}f$ doesn't exist because $1$ is not an accumulation point, then the function is not continuous at $1$ (Otherwise it should be $\lim_{x\to 1}f=f(1)$).

2) According to this answer , as far as I can understand a function is continuous at an isolated point.

I don't understand.

Edit:

• Spivak's definition of limit: The function $f$ approaches to $l$ near $a$ means $\forall \epsilon > 0 \; \exists \delta > 0 \; \forall x \; [0<|x-a|<\delta\implies |f(x)-l|<\epsilon]$

• Spivak's definition of continuity: The function $f$ is continuous at $a$ if $\lim_{x\to a}f(x)=f(a)$