# $C^{1}(\mathbb{R})$ be the collection of continuously differentiable functions on $\mathbb{R}$.

I came across the following problem:

Let $C^{1}(\mathbb{R})$ be the collection of continuously differentiable functions on $\mathbb{R}$.Let $S$=$\{f \in C^{1}(\mathbb{R}):f(0)=0,f(1)=1,|f'(x)|\leqslant 3/4, \forall x\in \mathbb{R} \}.$Then which of the following option is correct?

(a)S is empty, (b)S is non-empty and finite, (c)S is countably infinite, (d)S is uncountable.

According to the mean value theorem there must be a point $x\in(0,1)$ where $$f'(x) = \frac{f(1) - f(0)}{1-0} = 1$$ That should answer your question.
Thanks a lot.I have got it. Since in $S$,$|f'(x)|\leqslant 3/4$ whereas $|f'(x)|=1$,$S$ must be empty.So $(a)$ will be the correct choice. –  learner Dec 13 '12 at 4:26