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 $f(x_1)=g(x_1).$


for b> $g(x)=f(x+1)-f(x)$ then $\exists c \in [0,1]$ s.t. $g(c)=0 \implies f(c)=f(c+1)$ .
for c> if $f(x) \neq g(x) $ then $r(x)=f(x)-g(x) \neq0 $ for all $x$.
$0 \neq r(f(x))+r(g(x))=f(f(x))-g(f(x))+f(g(x))-g(g(x))=f(f(x))-g(g(x))$ for all $x$ which is a contradiction so $f(x_1)=g(x_1)$ for some $x_1$
