Interpretation of Taylor's theorem

Question

The following is Taylor's theorem given in a numerical analysis textbook:

I couldn't interpret what $C^n[a,b]$ means and what the intuition behind it is. Could someone explain?

Answer 1

The notation $C^n[a,b]$ is referring to a differentiability class:

In general, the classes $C^k$ can be defined recursively by declaring $C^0$ to be the set of all continuous functions and declaring $C^k$ for any positive integer $k$ to be the set of all differentiable functions whose derivative is in $C^{k−1}$.

Basically, $C^0$ denotes a class of functions that are continuous. Next, $C^1$ consists of functions that are differentiable with a continuous first derivative, and etc.

Essentially then, $C^n$ denotes the set of all $n$-differentiable functions, i.e. a function that differentiable $n$ times. The interval $[a,b]$ means that is differentiable $n$ times over the interval $[a, b]$ with the derivative being continuous. Thus, $f', f'', f''' \dots f^{(n)}$ exist on $[a,b]$ and are continuous.

The notation $f \in C^n[a,b]$ means that $f$ is in the set $C^n$ on $[a, b]$, thus has $n$ derivatives on the interval $[a, b]$. This is also basically explained by the next few words:

Suppose $f \in C^n[a,b]$, that $f^{(n+1)}$ exists on $[a,b]$ …

This implies that $f^{(n)}$ exists and is continuous because it at least must be because continuity is a requirement of differentiability. The fact $f^{(n+1)}$ is needed to exist is because it is needed for the remainder term of Taylor's Theorem.