I am going by what I am seeing on Wikipedia: In 1D, the standard definition for differentiability is,
A function $f:U\to\mathbb{R}$, defined on an open set $U\subset\mathbb{R}$, is said to be ''differentiable'' at $a\in U$ if the derivative $$f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$ exists.
Other times, $a$ is defined to be a point which is in the interior of a not necessarily open domain.
However, when I read some other references such as Terence Tao's Analysis I, this assumption does not seem to exist and I don't think it is a mistake or an omission.
Can someone help me understand whether openness and interiority are necessary in order to define differentiability?