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I am reading a mathematics textbook on the subject of numerical analysis. In one theory the author says let us assume $f$ to be a function in $C^{n+1}[a,b]$.

I understand that $[a, b]$ is the interval limits. What I don't understand is "what the letter $C$ stand for".

It then continues to read, let $p$ be the the polynomial of degree at most $n$ that interpolates $f$ at $n+1$ distinct points.

With the above context what does $n+1$ in $C^{n+1}$ mean?

Dr. Ehsan Ali
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    It means $f$ is a function defined on $[a, b]$ that is continuously differentiable $n+1$ times. – R_D Apr 29 '16 at 07:26

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Your textbook is Numerical Analysis: Mathematics of Scientific Computing, correct? This notation is introduced in Chapter 1 "Mathematical Preliminaries", Section 1.1 "Basic Concepts and Taylor's Theorem", page 6 in the third edition.

Chris Culter
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  • Oh yes! I can see the definition there. I suspected that the author might have had defined it in previous sections, but I could not find it in first glance. – Dr. Ehsan Ali Apr 29 '16 at 08:18
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$C^n[a,b]$ is the set / space of $n$ times continuously differentiable functions on the interval $[a,b]$. Including $C^0$, the continuous functions, and $C^\infty$, the smooth functions.

Arthur
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Taken from wikipedia: https://en.wikipedia.org/wiki/Smoothness#Differentiability_classes

A (n+1) class function means that the (n+1) derivative (and all the previous ones) of the function does exist and is continuous.

jasev
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