What does C mean when talking about range of functions?

Question

I am looking at the definition of the generalisation of rolle's Theorem:

• if $f\in C^{n-1}[a,b]$
• if $f(x)$ is $n$ times differentiable for $a<x<b$
• if $f(x_k) = c$ for $x_0=a < x_1 < x_2 < \cdots < x_n=b$

But what does $C^{n-1}$ mean in this context?

Answer 1

A function is said to be of a class $C^k$ if for all integers $n \leq k$ the derivatives $f^{(n)}(x)$ are continuous and exist. If a function is infinitely differentiable, for all $n >0$, it is of class $C^{\infty}$.

Moreover, one can specify it is of a class of differentiability over only some subset. For example, if I want to say that $f(x)$ is a smooth function in $[0,1]$, we may write $C^\infty([0,1])$.

The notation without brackets is also common as in your post, that is, e.g., $C^\infty [0,1]$. Furthermore, it is common to see $C^\infty$ used as a synonym for smooth, e.g. a $C^\infty$-manifold.