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Let $\mathcal{C}^1(\mathbb{R})$ denote the set of all continuously differentiable real valued functions defined on the real line. Define $$A=\{f\in\mathcal{C}^1(\mathbb{R})\mid f(0)=0,\,f(1)=1,\,|f'(x)|\leq 1/2\,\text{ for all }\,x\in\mathbb{R}\}$$ where $f'$ denotes the derivative of the function $f$. Pick out the true statement:

(a) $A$ is an empty set.
(b) $A$ is a finite and non-empty set.
(c) $A$ is an infinite set.

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I don't know which result from analysis I should apply here, please give me some hints or remark

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Mean value theorem. – Martin Jan 4 '13 at 5:29
Yeah, I unthinkingly overstepped "hints or remark." – Cam McLeman Jan 4 '13 at 5:38
up vote 2 down vote accepted

By the mean value theorem, if $f(0)=0$ and $f(1)=1$, then there exists $c\in(0,1)$ such that $$ f'(c)=\frac{f(1)-f(0)}{1-0}=1, $$ which is impossible if $|f'(x)|\leq \frac{1}{2}$ for all $x\in\mathbb{R}$. So the set is empty.

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