continuity - Problem of continuous real valued function

Which ofthe following statements are true?

a.If $f:\mathbb R\to\mathbb R$ is injective and continuous, then it is strictly monotonic.

b.If $f\in C[0,2]$ is such that $f(0)=f(2)$,then there exists $x_1,x_2$ in [0,2] such that $x_1-x_2=1$ and $f(x_1)=f(x_2).$

c.Let $f$ and $g$ be continuous real valued function on $\mathbb R$ such that for all $x\in \mathbb R$ wehave $f(g(x))=g(f(x.))$ If there exists $x_0\in \mathbb R $ such that $f(f(x_0))=g(g(x_0))$ then there exists $x_1\in\mathbb R$ such that $f(x_1)=g(x_1).$