Calculus Small Changes, Big Ideas

Section 23.1 Differentiability classes

• Every differentiable function is continuous.

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• A function is continuously differentiable if it is differentiable and its derivative is continuous.

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• A function is of differentiability class \(C^{k}\) if the derivatives \(f',f'',\cdots,f^{(k)}\) exist and are continuous.

  • \(C^{0}\text{:}\) continuous functions

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  • \(C^{1}\text{:}\) continuously differentiable functions

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  • \(C^{\infty}\text{:}\) smooth functions

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  • \(C^{\omega}\text{:}\) analytic functions. (May not put this here in the end.)

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    An example of a function that is smooth but not analytic is the bump function

    \begin{equation*} f(x)=\begin{cases}\ee^{-\frac{1}{1-x^2}}&\text{if $|x|<1$}\\ 0&\text{otherwise}\end{cases} \end{equation*}

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• Often interesting example:

  \begin{equation*} g_{n}(x)=\begin{cases}x^{n}\sin(1/x)&\text{if $x\neq 0$}\\ 0&\text{if $x= 0$}\end{cases} \end{equation*}

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• Remind about Intermediate Value Theorem, which can now be proven using known tools

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• Darboux’s theorem: Suppose \(f:[a,b]\to\bb R\) is differentiable. If \(\alpha\) is between \(f'(a)\) and \(f'(b)\text{,}\) there exists \(c\in(a,b)\) such that \(f'(c)=\alpha\text{.}\)

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