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Can anyone give examples of a few functions that belong to $\mathcal C^1[0,1]$ and some functions that do not belong there?

It’s the set of all continuous functions on $[0,1]$ which are continuously differentiable on $(0,1)$ where the derived functions has continuous extension on $[0,1]$.

It comes in the context of a problem that asks me to show that $\mathcal C^1[0,1]$ with a certain norm is normed linear space.

But before starting I want to get idea of its members.

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For instance, $f(x)=x$ belongs to $C^1\bigl([0,1]\bigr)$, because $f'(x)=1$. On the other hand, $s(x)=\sqrt x$ does not belong to $C^1\bigl([0,1]\bigr)$, beacuse, when $x\in(0,1]$, $s'(x)=\frac1{2\sqrt x}$, and you cannot extend it to a continuous function of $[0,1]$, since $\lim_{x\to0^+}s'(x)$ does not exist (in $\mathbb R$).

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