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.
I do not know how to progress. Please help.Thanks in advance for your time.