I've been going through Spivak's calculus and in it spivak states that for a function $f$ to be continuous on a closed interval $[a,b]$ the function is said to be continuous at the end points if $\lim_\limits{x\to{a^+}}f(x)=f(a)$
And
$\lim_\limits{x\to{b^-}}f(x)=b$
However can the same reasoning apply to the derivative of a function i know that limits are based off taking limits from any direction within the domain of the function that is $x$ lies in the domain of $f$ so for example if I had the function
$f:[a,b]\longrightarrow \mathbb{R}$
when calculating the derivative at the end point for example at b since f is not defined for points greater than b is it true to say that the derivative of the function at b is the left handed derivative of b
That is
$$
f'(b)=\lim_{h\to 0^-}\frac{f(b+h)-f(b)}{h}
$$
Thanks in advance.
Best Answer
You are correct, but you also don't need a provisional definitions when considering limits. The rigorous definition of a limit for a function $f:A\subseteq \mathbb{R}\to\mathbb{R}$ at a point $x_0$ only consider the elements of the domain. That is we call $L=\lim_{x\to x_0}f(x)$ if for every $\epsilon > 0$ there is a $\delta >0$ such that if $x\in A$ and $|x-x_0|<\delta$ then $|f(x)-f(x_0)|<\epsilon$. That means you only have to consider how the function approaches for $x$ close $x_0$ in the domain, so in the boundary case, the limit and the right-handed limit are the same as the elements above or below these limit points are not in the domain. Some people may have issues with this definition as it would imply all functions defined on, let's say, the integers are continuous. But further down the line you will see this is consistent with the notion of the subspace topology of a given topological space.