For $\displaystyle x\in[0,1]$ and $\displaystyle 0 \leq f(x)\leq x$ EDIT: and f(0)=0 and f(1)=1

Is it possible for f to be convex but not increasing?

And,

Is it possible for f to be increasing but not convex?

I'm pretty sure the first question is a "no", but how about the second one? If the answer is yes could you please provide an example?

Thanks