Answer
See answer below
Work Step by Step
Let $S$ be the set of function $f:[0,1] \rightarrow [0,1]$ such that $f(x) \leq x$ for all $x \in [0,1]$
Assume that $f:[0,1] \rightarrow [0,1]$ such that $f(x) = x$ for all $x \in [0,1]$. Then we have $f \in S$.
and $2f:[0,1] \rightarrow [0,1]$ such that $f(x) = x$ for all $x \in [0,1]$. Then $2f(1)=2\times1=2$ which gives us that $2f \notin S$ since $2 \notin [0,1]$
Thus $S$ is not a vector space over $R$
