Definition of derivative of real function in baby Rudin Let $f$ be defined (and real-valued) on $[a,b]$. For any $x\in [a,b]$ form a quotient $$\phi(t)=\dfrac{f(t)-f(x)}{t-x} \quad (a<t<b, t\neq x),$$ and define $$f'(x)=\lim \limits_{t\to x}\phi(t),$$ provided this limit exists in accordance with Defintion 4.1. 
I have one question. Why Rudin considers $t\in (a,b)$? What would be if $t\in [a,b]?$

In short, this is an example of Rudin's sublime succinctness. His definition includes the usual two-sided definition when $x \in (a,b)$ and also the one-sided definition $x = a$ or $x = b$. To get the one-sided definition, notice that he requires $t \in (a,b)$. 
Rudin was perhaps over-succinct in that he failed to mention that when $x = a$ or $x = b$, you have to interpret the limit as being one-sided. (Does Thm 4.1 include this?)
