I'm looking at Rudin's definition of the derivative. Here it is:
Let $f:[a,b]\to\mathbb{R}$. For any $x\in[a,b]$ form the quotient $\phi(t)=\frac{f(t)-f(x)}{t-x}$ where $a<t<b$ and $t\neq x$. Define $f'(x)=\lim_{t\to x}\phi(t)$ provided this limit exists. $f'$ is called the derivative of $f$. If $f'$ is defined at a point $x$, we say $f$ is differentiable at $x$.
So we can only say "$f$ is differentiable at $x$" if $f$ meets the requirements of the definition, one of which is that the domain of $f$ is some interval $[a,b]$. So $f$ having domain $[a,b]$ is necessary for $f$ to be differentiable at $x\in[a,b]$.
So then what does it mean for a function to be "differentiable everywhere" or "differentiable at every point of $\mathbb{R}$"? If we want to say a function $g$ has a derivative at every point of $\mathbb{R}$, then we cannot be using this definition since $g$ would necessarily not have domain $[a,b]$. We would only be able to say that some restriction of $g$ to an interval $I$ is differentiable at every point of $E\subseteq I$. (And this restriction is not $g$.)